A  DISSERTATION  ON  THE  DEVELOPMENT 

OF  THE 

SCIENCE  OF  MECHANICS 


BEING  A  STUDY  OF  THE   CHIEF  CONTRIBUTIONS  OF  ITS 
EMINENT  MASTERS,  WITH  A  CRITIQUE  OF  THE  FUN- 
DAMENTAL MECHANICAL  CONCEPTS,  AND  A 
BIBLIOGRAPHY  OF  THE  SCIENCE 


EMBODYING  RESEARCH  SUBMITTED  IN  PARTIAL  FULFILLMENT  OF 

THE  REQUIREMENTS   FOR  THE  DEGREE  OF  DOCTOR  OF 

SCIENCE   IN  NEW   YORK  UNIVERSITY,  1908 


BY 

DAVID  HEYDORN   RAY 

BACHELOR  OF  ARTS,  COLLEGE   OF   CITY   OF  NEW  YORK;    BACHELOR  OF  SCIENCE  AND   MASTER   OF 

ARTS,    COLUMBIA   UNIVERSITY;     CIVIL   ENGINEER,    NEW  YORK   UNIVERSITY;     INSTRUCTOR 

IN  THE   COLLEGE   OF  THE   CITY  OF   NEW   YORK;     CONSULTING   ENGINEER 


»,      1     »        0 


LANCASTER,  PA. 


\\^ 


CONTENTS. 


INTRODUCTION. 

NATURAL   SCIENCE. 

PART   I. 

BEGINNINGS  IN  MECHANICS. 

The  Period  of  Antiquity,  iogoo  B.  C.  to  500  A.  D. 

PAGE. 

1.  The  Science  of  Mechanics 8 

2.  Science  in  Antiquity 12 

3.  Archimedes 19 

PART   II. 
The  Medieval  Period,  500  A.  D.  to  1500  A.  D. 

1.  The  Medieval  Attitude  toward  Science 33 

2.  The  Influence  of  Arabian  Culture 39 

3.  The  Period  of  the  Renaissance 42 

4.  The  Contribution  of  Stevinus 45 

5.  The  Contribution  of  Galileo 52 

PART  III. 

MODERN  MECHANICS. 

The  Modern  Period,  1500  to  1900. 

1.  Characteristics  of  the  Modern  Period 60 

Huygens 63 

2.  Newton 69 

3.  The  Contributions  of  Varignon,  Leibnitz,  the 

Bernoullis,  Euler  and  D'Alembert 77 

4.  The  Contributions  OF  Lagrange  AND  Laplace.  . .   106 

iii 


iv  contents. 

5.  Recent  Contributions.    The  Law  of  Conserva- 

tion    117 

6.  The  Ether.    Energy.     Dissociation  of  Matter.  125 

PART   IV. 

CONCLUSION. 

1.  Conclusions  and  Critique  of  the  Fundamental 

Concepts  of  the  Science 133 

2.  Tabular   View   of   the    Development   of    Me- 

chanics    134 

3.  Bibliography 146 


INTRODUCTORY  CHAPTER. 
NATURAL  SCIENCE. 

The  word  mechanics,  though  it  indicated  of  old  the  study 
of  machines,  has  long  since  outgrown  this  limited  meaning 
and  now  embraces  the  entire  study  of  moving  bodies,  both 
large  and  small,  suns  and  satellites,  as  well  as  atoms  and  mole- 
cules. The  phenomena  of  nature  present  to  us  a  world  of 
change  through  ceaseless  motion.  Mechanics  is  the  "Science 
of  Motion"  as  the  physicist  Kirchhoff  has  defined  it,  and  has 
all  natural  phenomena  for  its  field  of  investigation.  Why 
things  happen  and  how  they  happen  are  the  questions  that 
here  present  themselves. 

It  was  a  long  time  before  the  distinction  between  "why" 
and  "how"  was  drawn,  but  when  once  the  question  "why" 
was  turned  over  to  the  metaphysician  and  the  theologian, 
and  attention  was  concentrated  on  "Aow,"  then  mechanics 
made  progress.  Men  then  began  to  discover  "how  things 
go,"  and  to  try  their  hand  at  invention. 

It  is  not  the  purpose  here  to  touch  upon  either  the  meta- 
physical or  the  psychological  aspect  of  phenomena,  nor  the 
mystery  of  vegetable  or  animal  activities,  but  to  trace  the 
development  of  Mechanics  as  a  science  from  the  earliest  records 
to  the  present  time,  first  analyzing  the  contributions  made  to 
it,  step  by  step,  and  then  touching  upon  their  use  and  value. 

As  the  French  philosopher  Comte  first  noted,  three  stages 
are  apparent  in  the  growth  of  human  knowledge.  In  the 
first  stage,  man  ascribed  every  act  to  the  direct  interposition 
of  the  Deity,  in  the  second  he  tried  to  analyze  the  Deity's 
motives  and  so  tried  to  learn  "why,"  while  in  the  third,  men 
came  to  regard  the  inquiry  "why"  as  profitless  and  ask  "how." 
In  this  last  stage,  they  accept  the  universe  and  are  content 
with  learning  all  they  can  of  how  it  goes.  With  this  last 
attitude,  called  positivism,  science  flourishes.  Out  of  it  grew 
the  notion  of  utilitarianism, — the  devotion  of  all   energies 


2  THE  SCIENCE  OF  MECHANICS. 

toward  the  improvement  of  the  conditions  of  life  on  earth. 
Though  this  later  philosophy  cannot  entirely  justify  itself, 
it  is  commonly  identified  with  the  scientific  attitude  of  mind. 

By  the  long  road  of  experience,  by  blunder,  trial  and  experi- 
ment, men  first  gathered,  it  seems,  ideas  of  things  that  appear 
always  to  happen  together  as  by  a  necessary  sequence  of 
"cause  and  effect."  Of  the  stream  of  appearances  continu- 
ously presenting  themselves,  some  are  invariably  bound  to- 
gether, being  either  simultaneous  or  successive,  the  presence 
or  absence  of  the  others  apparently  making  no  difference. 
Those  having  no  influence  may  reasonably  be  ignored  and 
eliminated  as  of  no  consequence.  In  this  way,  the  method 
of  abstracting  from  the  great  multitude  of  phenomena  those 
that  are  mutually  dependent  seems  to  have  been  evolved. 

Barbarous  peoples  do  not  possess  a  clear  notion  of  sequence 
or  of  the  interdependence  of  things.  They  are  prone  to  regard 
the  consequence  of  an  action  as  accessory,  as  something  done 
by  an  invisible  being  or  a  god.  An  action  is  performed  by 
them,  and  what  is  commonly  called  by  us  the  result  is  con- 
ceived by  them  as  the  simultaneous  act  of  their  god.  Their 
medicineman  is  thought  of,  as  one  proficient  in  the  art  of 
appealing  to  the  moods  and  whims  of  their  gods  propitiously. 
Even  the  Greeks  and  Romans,  the  founders  of  our  European 
civilization,  were  accustomed  to  be  guided  in  affairs  of  state 
and  of  the  home  by  omens,  by  the  flight  of  birds,  and  the 
inspection  of  the  entrails  of  animals, — most  naive  examples 
of  traditional  error  in  the  interdependence  of  simultaneous 
phenomena. 

Things  which  we  now  understand  to  have  not  the  slightest 
relation  with  each  other  were  systematically  confounded  by 
the  ancients.  For  thousands  of  years  belief  in  astrology  was 
general  in  Europe  and  the  universahty  of  the  belief  is  at- 
tested by  such  words  as  ill-starred,  disastrous,  consider  and 
saturnine,  all  of  which  are  manifestly  of  astrological  ety- 
mology. It  was  only  very  slowly  and  gradually,  step  by  step, 
that  men  came  to  think  of  phenomena  quantitatively  rather 
than  qualitatively,  and  to  arrive  at  a  more  rational  concep- 
tion of  nature  through  experience  and  reflection. 


NATURAL   SCIENCE.  3 

As  the  interrelation  of  things  came  to  be  more  clearly  per- 
ceived, people  began  to  say  they  could  "explain  things," 
meaning  that  they  had  arrived  at  a  familiarity  with,  and  had 
begun  to  recognize  certain  permanent  elements  and  sequences 
in  the  variety  of  phenomena.  By  joining  these  elements,  they 
constructed  a  chain  and  attained  to  a  more  or  less  extensive 
and  consistent  comprehension  of  the  relations  of  phenomena 
by  a  co-ordination  of  their  permanent  elements. 

If  these  elements  are  linked  together  logically,  the  satis- 
factoriness  of  "the  explanation"  depends  upon  the  length  of 
the  chain.  The  longer  the  chain,  the  further  it  reaches,  and 
the  more  satisfied  one  is,  the  more  one  "understands"  the 
matter.  This  is  the  general  method  of  "learning  things,"  and 
the  information  so  collected  may  be  called,  as  Prof.  Karl 
Pearson  has  called  it,  an  "intellectual  resume  of  experience." 
But  it  should  be  noted  that  it  is  rarely  the  simple  correlation  of 
things  that  will  stand  the  test  of  experiment. 

There  is  in  this  method  abundant  chance  to  go  wrong.  It  is 
difficult,  and  especially  troublesome  for  a  beginner,  untrained 
in  this  process,  to  decide  what  things  really  do  not  have  effect 
and  hence  may  be  excluded  from  consideration.  And  if  it  is 
difficult  for  the  beginner  in  science  to-day,  surely  it  was  im- 
mensely more  so  for  primitive  men.  Students  are  wont  to 
complain  of  the  artificiality  of  geometry  and  mechanics.  Fac- 
tors which  they  feel  do  make  a  difference  in  reality  do  not 
seem  to  them  to  be  fully  allowed  for,  or  they  are  troubled  by  a 
feeling  of  uncertainty  as  to  the  equity  of  the  allowance.  The 
peculiar  value  of  mathematical  studies  lies  just  here  in  the 
rigorous  training  in  reasoning.  Whatever  a  student's  success 
with  his  mathematics,  few  make  its  acquaintance  without 
receiving  wholesome  lessons  of  patient  application  of  the  in- 
tellectual method  by  which  mankind  has  won  its  mastery 
over  natural  forces. 

We  may  quote  here  to  advantage  Prof.  Faraday.^  "There 
are  multitudes  who  think  themselves  competent  to  decide, 
after  the  most  cursory  observation,  upon  the  cause  of  this  or 

1  Lecture  delivered  before  Royal  Institution  of  Great  Britain, — "On  Edu- 
cation of  the  Judgment." 


4  THE   SCIENCE   OF  MECHANICS. 

that  event,  (and  they  may  be  really  very  acute  and  correct 
in  things  familiar  to  them) : — a  not  unusual  phrase  with  them 
is,  that  'it  stands  to  reason,'  that  the  effect  they  expect  should 
result  from  the  cause  they  assign  to  it,  and  yet  it  is  very  dif- 
ficult, in  numerous  cases  that  appear  plain,  to  show  this  reason, 
or  to  deduce  the  true  and  only  rational  relation,  of  cause  and 
effect. 

"If  we  are  subject  to  mistake  in  the  interpretation  of  our 
mere  sense  impressions,  we  are  much  more  liable  to  error 
when  we  proceed  to  deduce  from  these  impressions  (as  sup- 
plied to  us  by  our  ordinary  experience),  the  relation  of  cause 
and  effect;  and  the  accuracy  of  our  judgment,  consequently, 
is  more  endangered.  Then  our  dependence  should  be  upon 
carefully  observed  facts,  and  the  laws  of  nature;  and  I  shall 
proceed  to  a  further  illustration  of  the  mental  deficiency  I 
speak  of,  by  a  brief  reference  to  one  of  these. 

"The  laws  of  nature,  as  we  understand  them,  are  the  founda- 
tion of  our  knowledge  in  natural  things.  So  much  as  we 
know  of  them  has  been  developed  by  the  successive  energies 
of  the  highest  intellects,  exerted  through  many  ages.  After 
a  most  rigid  and  scrutinizing  examination  upon  principle  and 
trial,  a  definite  expression  has  been  given  to  them;  they  have 
become,  as  it  were,  our  belief  or  trust.  From  day  to  day  we 
still  examine  and  test  our  expression  of  them.  We  have  no 
interest  in  their  retention  if  erroneous;  on  the  contrary,  the 
greatest  discovery  a  man  could  make  would  be  to  prove  that 
one  of  these  accepted  laws  was  erroneous,  and  his  greatest 
honour  would  be  the  discovery.  .   .   . 

"These  laws  are  numerous,  and  are  more  or  less  compre- 
hensive. 'They  are  also  precise;  for  a  law  may  present  an 
apparent  exception,  and  yet  not  be  less  a  law  to  us,  when 
the  exception  is  included  in  the  expression.  Thus,  that  eleva- 
tion of  temperature  expands  all  bodies  is  a  well-defined  law, 
though  there  be  an  exception  in  water  for  a  limited  tempera- 
ture; we  are  careful,  whilst  stating  the  law  to  state  the  excep- 
tion and  its  limits.  Pre-erriinent  among  these  laws,  because 
of  its  simplicity,  its  universality,  and  its  undeviating  truth, 
stands  that  enunciated  by  Newton  (commonly  called  the  law 


NATURAL    SCIENCE.  5 

of  gravitation),  that  matter  attracts  matter  with  a  force  in- 
versely as  the  square  of  the  distance.  Newton  showed  that, 
by  this  law,  the  general  condition  of  things  on  the  surface  of 
the  earth  is  governed;  and  the  globe  itself,  with  all  upon  it 
kept  together  as  a  whole.  He  demonstrated  that  the  motions 
of  the  planets  round  the  sun,  and  of  the  satellites  about  the 
planets,  were  subject  to  it.  During  and  since  his  time,  certain 
variations  in  the  movements  of  the  planets,  which  were  called 
irregularities,  and  might,  for  aught  that  was  then  known,  be 
due  to  some  cause  other  than  the  attraction  of  gravitation, 
were  found  to  be  its  necessary  consequences.  By  the  close 
and  scrutinizing  attention  of  minds  the  most  persevering  and 
careful,  it  was  ascertained  that  even  the  distant  stars  were 
subject  to  this  law;  and,  at  last,  to  place  as  it  were  the  seal 
of  assurance  to  its  never-failing  truth,  it  became,  in  the  minds 
of  Leverrier  and  Adams  (1845),  the  foreteller  and  the  dis- 
coverer of  an  orb  rolling  in  the  depths  of  space,  so  large  as 
to  equal  nearly  sixty  earths,  yet  so  far  away  as  to  be  invisible 
to  the  unassisted  eye.  What  truth,  beneath  that  of  revelation, 
can  have  an  assurance  stronger  than  this!" 

Such  is  the  process  of  scientific  induction.  It  was  by  linking 
ideas  together  in  an  orderly  way,  by  forming  and  verifying 
hypotheses,  that  men  finally  came  to  the  "principles,"  and 
"formulae,"  which  embody  these  general  "truths"  or  "laws  of 
nature."  In  this  way  knowledge  has  been  built  up,  chain  by 
chain,  into  a  more  or  less  complete  system  of  the  relations  of 
things.  Without  asking  the  "why"  of  it  all  one  can  see 
"how"  it  goes  together  by  running  along  the  chains  from  link 
to  link.  In  a  word  this  knowledge  is  relative,  and  therefore 
quantitative,-  and  that  is  why  numbers  and  mathematics  play 
so  large  a  part  in  the  exact  sciences,  and  in  mechanics. 

The  guiding  principle  in  all  this  is  the  belief  in  the  con- 
stancy of  the  order  of  nature  founded  on  the  experience  of 
the  human  race.  On  this  belief  are  based  all  scientific  calcu- 
lations and  deductions.  This  is  sometimes  formulated  as  a 
"Law  of  Causality,"  affirming  that  every  effect  has  a  sufficient 
cause  and  that  the  relation  of  cause  and  effect  is  one  of  in- 
variable sequence,  if  not  interfered  with  by  conditions  or 
circumstances  that  make  the  cases  dissimilar. 


6  THE    SCIENCE   OF   MECHANICS. 

Information  thus  systematized,  verified  and  formulated  into 
truths  or  general  principles  is  called  Natural  Philosophy  or 
Natural  Science.  The  Science  of  Mechanics  is  the  oldest  and 
one  of  the  most  important  divisions  of  Natural  Philosophy. 
This  knowledge  of  the  interdependence  and  inter-relation  of 
phenomena  makes  it  possible  to  "predict"  and  "control"  them, 
and  keeps  us  from  making  hasty  and  erroneous  inferences. 
When  developed  with  this  view,  applied  science  or  applied 
mechanics  is  the  usual  designation,  and  that  such  information 
is  power  to  one  who  has  the  skill  to  apply  it,  need  not  be  dwelt 
upon.  As  Herbert  Spencer  says  in  his  volume  on  Education:^ 
"On  the  application  of  rational  mechanics  depends  the  success 
of  nearly  all  modern  manufacture.  The  properties  of  the 
lever,  the  wheel  and  axle,  etc.,  are  involved  in  every  machine 
— every  machine  is  a  solidified  mechanical  theorem;  and  to 
machinery  in  these  times  we  owe  nearly  all  production.'' 
Elsewhere  he  says :  "All  Science  is  prevision ;  and  all  prevision 
ultimately  helps  us  in  greater  or  less  degree  to  achieve  the 
good  and  to  avoid  the  bad."^ 

It  is  not  the  intention  here  to  discuss  or  even  to  enumerate 
the  triumphs  in  the  practical  applications  of  mechanics.  The 
utilization  of  power,  of  the  strength  of  animals,  the  power 
of  the  wind,  of  waterfalls,  of  steam  and  of  electromagnetic 
attraction,  constitutes  the  art  of  machine  contrivance 
rather  than  the  science  of  mechanics.  Progress  in  theoretical 
mechanics  has  always  brought  in  its  train  an  advance  in 
machinery. 

The  innumerable  engines  for  enlightenment  and  destruction, 
the  cylinder-printing-press  and  the  machine-gun  which  have 
changed  and  are  altering  the  economic,  social  and  religious 
prospect  of  nations  and  tribes  are  the  direct  result  of  the 
application  of  the  principles  of  the  science  of  mechanics.  With 
further  advance  in  theory  and  systematic  experimentation  even 
more  revolutionizing  contrivances  will  inevitably  follow. 
When  invention  has  realized  the  theoretical  surmise  that  the 
"molecular  energy"  in  a  cup  of  tea  is  sufficient  to  tumble  down 

ip.  30. 

2"First  Principles,"  p.  15. 


NATURAL   SCIENCE.  7 

a  town,  we  may  expect  an  Age  of  Power  ushering  in  wonders 
untold.^ 

With  the  philosophy  that  denies  the  existence  of  reaUties 
outside  of  the  mind  we  shall  not  trouble  ourselves  here. 
Mechanics  regards  a  "truth"  or  a  "law"  not  as  subjective  but 
as  objective,  holding  that  an  external  world  exists  and  that 
truth  is  a  relation  of  conformity  between  the  mental  world 
of  perceptions  and  inferences,  and  really  existing  objects  and 
their  relations.  Unless  this  and  the  validity  of  the  principle 
of  logical  inference  be  conceded,  our  science  is  futile.  The 
mental  processes  by  which  the  victories  of  Science  are  won  are 
in  no  wise  different  from  those  used  by  all  in  daily  affairs.  As 
Huxley  says :  "Science  is  nothing  but  organized  common  sense. 
The  man  of  Science  simply  uses  with  scrupulous  exactness 
the  methods  which  we  all  habitually  and  at  every  moment, 
use  carelessly.  Nor  does  that  process  of  induction  and  de- 
duction by  which  a  lady,  finding  a  stain  of  a  peculiar  kind 
on  her  dress,  concludes  that  somebody  has  upset  the  inkstand 
thereon,  differ  in  any  way,  in  kind  from  that  by  which  Adams 
and  Leverrier  discovered  a  new  planet." 

Nevertheless  there  will  always  remain  certain  ultimate  truths 
which  cannot  be  proved  and  which  must  be  consideredas  axiom- 
atic and  intuitive.  This  should  not  invalidate  our  conclusions 
and  we  will  not  enter  upon  a  discussion  of  these  questions  here. 

The  science  of  mechanics  has  then,  for  its  subject  matter, 
the  motion-phenomena  of  the  universe.  Its  growth  is  co- 
extensive with  that  of  the  race,  and  one  of  its  functions  is  the 
widening  of  its  perceptions.  It  is  obviously  a  subject  of 
primary  importance,  for  from  apparent  chaos,  it  evolves  rules 
and  principles  of  practical  utility,  and  so  increases  knowledge 
and  efhciency,  and  consequently  happiness,  through  power 
and  dominion  over  nature. 

^Suppose  that  a  cup  of  tea  (about  lOO  cubic  centimeters)  could  be 
suddenly  and  completely  dissociated,  after  the  manner  of  the  radio-active 
emissions  of  radium,  into  a  cloud  of  particles  with  a  velocity  similar  to 
radium  emanations  of  say  100,000  kilometers  a  second  (about  one-third 
the  velocity  of  light),  then  a  simple  calculation  by  the  theoretical  formula 
for  energy,  J^twi)^,  gives  3^  X.1/9.8X  100, 000,000^  =  50,000,000,000,000 
kilogramme-meters,  equal  to  the  energy  of  explosion  of  about  500,000  tons 
of  rifle  powder,  or  enough  energy  to  drive  an  express  train  around  the  globe 
a  hundred  times. 


PART   I. 

I.     THE  SCIENCE  OF  MECHANICS. 

The  most  common  of  all  our  experiences  is  the  motion  of 
solid  bodies.  No  idea  is  more  frequently  with  us  than  the 
idea  of  such  movements.  It  seems  to  be  the  first  experience 
of  the  dawning  intellect  and  it  is  soon  fully  developed  by 
boyhood's  games  of  marbles  and  tops.  Indeed,  there  is 
nothing  that  our  imagination  pictures  with  greater  ease  and 
readiness,  than  a  moving  speck  or  particle.  There  is  there- 
fore considerable  satisfaction,  and  an  appealing  reasonableness 
and  inevitableness  in  the  idea  of  classifying  phenomena  on 
the  basis  of  this  familiar  experience. 

This  idea  and  another,  quite  as  familiar,  namely,  that  com- 
mon objects  can  be  crushed  and  broken  into  many  small  par- 
ticles and  ground  to  dust  so  small  as  to  seem  indivisible,  are 
fundamental,  and  upon  them  the  science  of  mechanics,  as  a 
scheme  of  motions  and  equilibrium  of  particles  has  been  built 
up.  Masses  either  change  their  relative  position  or  they  do 
not.  How  they  move,  rather  than  why  they  move,  is  the 
question  of  Mechanics.  It  is  especially  the  circumstances  of 
motion  or  of  rest  that  are  the  subject  of  investigation  of  the 
science. 

In  its  formal  presentation  in  textbooks,  Mechanics  is  now 
defined  by  an  American  Professor,  Wright,  as  "the  science 
of  matter,  motion,  and  force";  by  an  English  Professor,  Ran- 
kine,  as  the  "science  of  rest,  motion  and  force";  by  a  German 
Professor,  Mach,  as  that  branch  of  Science  which  is  "concerned 
with  the  motions  and  equilibrium  of  masses."  These  defini- 
tions do  not  differ  essentially. 

The  questions  at  once  present  themselves  what  is  force, 
what  is  matter,  what  is  mass?  Etymology  does  not  help  us. 
The  further  back  one  goes,  the  more  indistinctive  and  general 
is  the  idea  corresponding  to  a  scientific  term.  The  terms, 
matter,  mass,  force  and  weight  lose  precision  as  we  trace  them 


THE   SCIENCE   OF  MECHANICS.  9 

back.  Matter  leads  us  back  to  the  Latin,  materia,  i.  e., 
substance  for  construction  or  building.  Mass  appears  to  be 
derived  from  the  Greek  root  (Mdaaetv),  to  knead.  So  by 
derivation,  matter  means  the  substance  or  pith  of  a  body, 
and  mass  means  anything  kneaded  together  like  a  lump  of 
dough.  The  fundamental  idea  of  mass  is  then  an  agglutinated 
lump.  Weight  is  of  Saxon  derivation  from  a  root  meaning  to 
bear,  to  carry,  to  lift.  Force  appears  to  come  from  the  Latin 
root,  fortia,  meaning  muscular  vigor  and  strength  for  violence. 
It  is  an  anthropomorphic  concept,  and  is  suggestive  of  myth- 
ology in  its  application  to  inanimate  things. 

All  these  terms  are  derived  from  words  expressing  distinct 
muscular  sensations.  Here  in  the  last  analysis  we  come  back 
to  sense-impressions.  A  mass  is  an  agglutinated  lump  as  of 
kneaded  dough,  weight  is  resistance  to  lifting,  and  force  is  some- 
thing that  produces  results  analogous  to  those  produced  by 
muscular  exertion.  We  cannot  analyze  these  simple,  immediate 
perceptions,  nor  can  we  analyze  motion.  Motion  is  a  sense  of 
free,  unrestricted  muscular  action.  Muscular  action  impeded 
gives  us  our  sense  of  force.  Perhaps  our  primitive  perception 
of  force  was  muscular  action  under  restraint  or  not  accom- 
panied by  motion.  From  these  sense-impressions  we  attain, 
by  inference,  the  idea  of  space,  i.  e.,  room  to  move  in,  and  the 
notion  of  time  or  uniformity  of  sequence.  Mechanics  might  ^; 
then  be  crudely  defined  as  a  scheme  of  the  relations  of  lumps 
of  matter  acted  upon  by  muscular  exertion  or  by  anything  ; 
that  produces  like  effects.  ) 

Observe  that  we  are  conscious  of  these  sense-impressions, 
comparatively  only.  We  are  aware  of  them  only  through 
change  in  their  intensity.  Here  in  our  endeavors  to  com- 
prehend and  to  define  the  ultimate  elements  of  mechanics  we 
have  borne  in  upon  us  the  relativity  of  knowledge.  The  con- 
viction that  the  human  intelligence  is  incapable  of  absolute 
knowledge  is  the  one  idea  upon  which  philosophers,  scientists, 
and  theologians  are  in  accord.  It  is  a  characteristic  of  con- 
sciousness that  it  is  only  possible  in  the  form  of  a  relation. 

"Thinking  is  relationing  and  no  thought  can  express  more 
than  relations,"  says  Herbert  Spencer  in  his  Chapter  on  the 


10  THE   SCIENCE   OF  MECHANICS. 

Relativity  of  Knowledge.  And  he  concludes:  "Deep  down  in 
the  very  nature  of  Life,  the  relativity  of  our  knowledge  is  dis- 
cernible. The  analysis  of  vital  actions  in  general,  leads  not 
only  to  the  conclusion  that  things  in  themselves  cannot  be 
known  to  us,  but  also  to  the  conclusion  that  knowledge  of 
them,  were  it  possible,  would  be  useless."^ 

But  though  we  are  limited  in  this  way  we  have  a  large  field 
in  the  building  of  a  scheme  of  inter-relations  of  the  relations 
which  comprise  our  conscious  perceptions.  This  is  the  purpose 
of  our  science  of  mechanics.  In  general  it  endeavors  to  inter- 
pret for  us  the  complex  relativity  of  phenomena  in  terms  of 
the  most  common  and  simplest  of  our  experiences,  namely 
the  relativity  of  motion  of  a  particle  and  the  relativity  of 
the  divided  parts  of  bodies. 

As  science  progresses  the  ideas,  mental  pictures,  and  terms 
found  serviceable  in  the  earlier  stages  are  bound  to  prove 
inadequate  later.  The  process  of  reorganizing  these  ideas, 
and  perfecting  terminology  is  slow,  but  in  it  there  is  unmistak- 
able evolutionary  progress. 

As  the  philologist  Nietzsche  says,  "Wherever  primitive  man 
put  up  a  word,  he  believed  he  had  made  a  discovery.  How 
utterly  mistaken  he  really  was!  He  had  touched  a  problem, 
and  while  supposing  he  had  solved  it,  he  had  created  an 
obstacle  to  its  solution.  Now,  with  every  new  knowledge,  we 
stumble  over  flint-like  petrified  words. "^ 

The  prehistoric  races  probably  explained  phenomena  by 
associating  with  everything  that  produces  motion,  some  in- 
visible god  whose  muscular  strength  was  the  force  of  wind, 
wave  or  waterfall.  We  find  in  all  languages,  survivals  of  this 
in  the  genders  ascribed  to  things  inanimate.  Indeed,  one  can 
dig  out  of  philology  and  mythology  a  petrified  primitive  natu- 
ral philosophy. 

To-day  we  sometimes  hear  that  all  phenomena  of  the  material 
world  are  explainable,  in  terms  of  matter,  motion,  and  force, 
or  by  the  whirl  of  molecules.  One  may  endeavor  to  make 
this  a  truism  by  defining  matter  as  anything  that  occupies 

^Spencer,  "First  Principles,"  Chapter  IV. 
''Nietzsche,  "Morgenrote,"  vol.  i,  47. 


THE   SCIENCE   OF  MECHANICS.  II 

space,  and  by  defining  force  as  any  agent  which  changes  the 
relative  condition  as  to  rest  or  motion  between  two  bodies, 
or  which  tends  to  change  any  physical  relation  between  them, 
whether  mechanical,  thermal,  chemical,  electrical,  magnetic, 
or  of  any  other  kind.  But  here  one  does  not  say  what  force 
is,  nor  what  matter  is.  The  chain  hangs  in  the  air;  it  does 
not  begin  or  end  anywhere,  but  the  relation  of  the  links  is 
apparent  and  serviceable.  Indeed,  the  idea  of  force  is  still 
fundamentally  the  same,  it  is  still  an  agent,  as  was  the  ancient 
nature-god,  though  much  less  definite,  nor  does  it  help  matters 
to  subdivide  force  and  mass. 

The  idea  of  force  as  a  latent  unknown  cause  is  a  historical  "-^ 
survival  of  our  primitive  conceptions  and  undergoes  trans- 
formation with  the  idea  of  force  as  a  "circumstance  of  motion,"    { 
which  was  developed  about  the  year  1700.     It  is  now  held    \ 
by  some  that  force  is  a  purely  subjective  conception.     For 
example,  Tait  says  in  his  "Newton's  Laws  of  Motion":  "We 
have  absolutely  no  reason  for  looking  upon  force  as  a  term 
for  anything  objective;  we  can,  if  we  choose,  entirely  dispense 
with  the  use  of  it.     But  we  continue  to  employ  it;  partly 
because  of  its  undoubted  convenience,  mainly  because  it  is 
essentially  involved  in  the  terminology  of  Newton's  Laws  of 
Motion,  which  still  form  the  simplest  foundation  of  our  subject. 
It  must  be  remembered  that  even  in  strict  science  we  use  such 
obvious  anthropomorphisms  as  the  'sun  rises,'  'the  wind  blows,' 
etc." 

Yet  though  there  may  be  no  such  reality  as  force,  mechanics  ^ 
will  probably  long  continue  to  be  known  as  the  dictionary 
defines  it,  as  "the  science  which  treats  of  the  action  of  forces 
on  bodies,  whether  solid,  liquid  or  gaseous."  We  do  not 
disparage  the  use  of  the  idea  and  term  force;  we  shall  have 
occasion  to  use  them  often.  But  it  should  be  noted  that  an 
evolution  in  terminology  is  involved  in  the  evolution  of 
science. 

Such  changes  in  conception  and  in  terminology  are  inevi- 
table. They  are  essential  characteristics  of  progressive  science 
which  seeks  continually  to  improve  the  definiteness  of  relation 
between  phenomena  by  making  clearer  vague  connections,  or 


12  THE   SCIENCE   OF   MECHANICS. 

by  discovering  new  relations.  The  relations  formerly  classed 
as  acoustic,  luminous,  thermal,  electric,  magnetic  and  chem- 
ical expressing  certain  constant  connections  of  antecedents 
and  consequents  are  now  generally  expressible  with  exactness 
in  the  terms  of  the  science  of  mechanics  which  is  built  on  the 
familiar  notions  of  motion  and  divisibility. 

As  a  matter  of  convenience,  the  science  has  come  to  be 
divided  into  Phoronomics  or  Kinematics,  the  study  of  pure 
motion  without  reference  to  the  nature  of  the  body  moved, 
or  how  the  motion  is  produced,  and  Dynamics,  the  "science 
of  force,"  "the  study  of  the  push  or  pull  of  bodies,"  or  "the 
science  of  the  properties  of  matter  in  motion."  It  is  evident 
that  in  some  cases  the  "forces  balance,"  giving  the  condition 
of  rest;  this  branch  of  the  study  is  called  Statics.  The  study 
of  unbalanced  forces  producing  motions  of  various  kinds  is 
called  Kinematics.  These  divisions  are  purely  arbitrary  and 
were  made  late  in  the  development  of  the  subject.  His- 
torically, the  study  of  Statics,  or  of  bodies  relatively  at  rest, 
was  the  first  to  be  undertaken  for  obvious  reasons. 

2.     THE  SCIENCE  OF  MECHANICS  IN  ANTIQUITY. 

It  is  the  verdict  of  conservative  geologists  and  physicists 
that  the  earth's  crust  is  at  least  25,000,000  years  old,  that 
period  of  time  being  required  for  the  deposit  of  the  depth  of 
about  50,000  feet^  of  sedimentary  rocks  that  research  discloses; 
and  it  is  the  opinion  of  conservative  authorities  that  rude  com- 
munities of  men  were  dwelling  in  the  broad  alluvial  valleys 
of  the  Nile,  Euphrates,  Ganges,  Hoang-Ho,  (perhaps  also  on 
the  ancient  Thames-Rhine  system),  as  early  as  25,000  years 
ago.  The  subsidence  of  these  broad  rivers  into  narrower 
channels  left  exposed  fertile  plains  in  their  old  bottoms  and 
islands  in  the  estuaries,  which  favored  the  development  of 
progressive  communities. 

This  was  particularly  true  of  the  Euphrates  valley  and  along 

the  Nile,  where  the  wild  wheat  and  barley  offered  food  and 

made  life  a  less  severe  struggle  for  existence.     Here  perhaps 

the  first  rude  camps  and  villages  were  developed.     But  even 

^130,000  feet  is  the  average  figure  suggested  by  Dr.  E.  Haeckel — p.  9 
"Evolution  of  Man,"  Vol.  2. 


THE   SCIENCE   OF  MECHANICS.  1 3 

these  early  communities  were  probably  in  possession  of  rude 
tools  and  weapons.  Darwin^  cites  instances  of  tools  used  by 
animals  and  we  must  imagine  that  even  the  very  earliest  com- 
munities of  men  were  acquainted  with  such  crude  mechanical 
appliances  as  the  lever  and  cord. 

The  researches  of  geologists  and  archaeologists  present  in- 
numerable stone  wedges,  flint  axes,  bone  and  horn  implements, 
and  primitive  tools  found  in  graves  of  the  stone  age,  or  on  the 
site  of  ancient  cave  and  lake  dwellings,  indicating  extensive 
mechanical  experience  in  prehistoric  times.^  An  instinctive 
familiarity  through  long  experience,  with  some  of  the  com- 
mon natural  processes,  and  a  knowledge  of  crude  cutting  and 
grinding  tools  must  then  be  accepted  as  very  ancient,  at  least 
twelve  or  fifteen  thousand  years  old. 

This  must  be  distinguished,  however,  from  a  mechanical 
theory  of  science,  which  is  the  product  of  reflection.  The 
latter  was  a  very  slow  and  gradual  evolution.  From  a 
long  experience  of  measuring  and  bartering,  a  knowledge 
of  numbers  probably  arose,  and  then  a  more  definite  knowl- 
edge of  the  simple  mechanical  devices  was  developed.  From 
these,  by  reflection  and  generalization,  rules  and  principles  were 
evolved.  In  the  ancient  Sanskrit  language  the  word  from 
which  "man"  comes,  appears  to  mean  to  estimate,  to  measure. 
Man  first  became  conscious  of  himself,  it  appears,  therefore,  as 
the  being  who  measures  and  weighs,  compares  and  reflects. 

Wedges,  pulleys,  windlasses,  oars  and  the  lever  in  various 
forms  were  used  before  any  rule  for  them  was  conceived  of; 
and  then  the  rules  for  centuries  remained  but  disjointed 
unrelated  statements  of  experience.  Only  very,  very  slowly 
were  they  mastered  and  made  into  a  body  of  mechanical 
knowledge.  As  this  process  proceeded,  the  fetishism  and 
mythology  invented  to  explain  natural  phenomena  declined 
before  a  more  rational  and  logical  group  of  mechanical  prin- 
ciples. But  traces  of  it  long  survived.  For  example,  the 
idea  that  "nature  abhors  a  vacuum"  is  a  late  survival  of  such 

iThe  Descent  of  Man,  Chapter  III,  "Tools  and  Weapons  used  by 
Animals." 

-Prehistoric  Times,  Sir  J.  Lubbock;  Ancient  Stone  Implements,  Evans; 
Man  and  the  Glacial  Period,  D.  F.  Wright;  Man's  Place  in  Nature,  T.  H. 
Huxley;  Origin  of  Species,  etc.,  C.  Darwin. 


14  THE   SCIENCE   OF  MECHANICS. 

fanciful  conceptions,  and  was  cited  as  late  as  1600  A.D.  But 
for  Science,  as  Spencer  says,  we  should  be  still  worshipping 
fetishes;  or  with  hecatombs  of  victims  be  propitiating  dia- 
bolical deities. 

It  seems  that  it  was  only  among  the  people  of  the  Eastern 
Mediterranean  coast  that  a  true  science  of  mechanics  was 
developed.  There  is  no  evidence  to  show  that  among  any 
of  the  peoples  of  the  Far  East  any  true  science  of  mechanics 
was  even  begun.  Indeed  some  of  the  people  of  the  yellow 
and  darker  races  still  live  in  the  stone  or  bronze  age.  Cer- 
tainly the  whole  development  of  the  science  as. we  have  it  is 
European.  Of  the  world's  population  of  1,500,000,000,  the 
200,000,000  of  Europe  and  the  100,000,000  of  America  who 
have  a  grasp  on  mechanical  science  are  in  control.  Half  of 
Asia's  700,000,000  are  held  subject  by  Europe's  Science,  and 
the  destiny  of  the  other  half  is  the  topic  of  the  hour. 

To  the  Babylonians  and  Phoenicians,  skilled  in  measuring, 
in  plane  surveying,  in  keeping  accounts,  and  in  seafaring, 
the  science  of  Europe  is  traced  back.  Centuries  before  the 
era  of  Greece,  the  Phoenicians  had  developed  a  crude  astron- 
omy and  were  practicing  and  slowly  improving  the  common 
mechanic  arts  and  trades.  They  are  not  to  be  credited  with 
originating  them  however,  for  scholars  have  traced  these  people 
back  to  a  mingling  of  tribes  of  primitive  Semetic  and  Aryan 
stock  which  took  place  in  the  Tigris-Euphrates  region  of  Asia, 
about  8000-10000  B.C. 

Here  a  remarkable  civilization  of  teeming  cities  had  devel- 
oped by  5000  B.C.  The  trials  and  troubles,  the  institutions, 
arts,  literature,  and  the  wail  of  the  prophets,  the  complete 
life  history  of  growth  and  decay  of  these  cities  may  be  read 
in  the  cuneiform  inscriptions  on  the  clay  tablets  in  the  British 
Museum.  With  the  shifting  of  the  trade  routes  to  the  north 
and  west,  through  the  Dardanelles,  their  prosperity  declined 
and  they  passed  out  of  existence. 

Perhaps  the  oldest  relic  of  their  mechanical  arts  is  the 
splendid  tablet  or  "stele"  set  up  in  the -temple  of  Lagash  by 
Eannatum  (c.  4200  B.C.).  One  side  shows  the  king  in  his 
chariot  leading  his  army  to  victory,  the  other  shows  the  wreck 


THE   SCIENCE   OF  MECHANICS.  1 5 

and  ruin  of  the  vanquished  whose  mangled  corpses  are  left 
to  the  vultures.  The  great  king  of  these  people,  Sargon  I 
(c.  3800  B.C.),  is  said  to  have  extended  his  conquests  west- 
ward as  far  as  the  Island  of  Cyprus,  the  land  of  copper. 
Bartering  expeditions  then  as  now  spread  information  and 
developed  the  arts  and  trades.  As  early  as  3000  B.C.  the 
Eg^'ptians  seem  to  have  become  a  power. 

It  seems,  then,  that  the  European  development  of  mechanics 
as  a  science  is  founded  on  at  least  3,000  or  4,000  years^  develop- 
ment of  the  recognized  mechanic  arts  and  trades,^  and  it  is 
probable  that  it  began  with  the  systematizing  of  craft  experi- 
ence and  the  formulation  of  this  experience  in  connection  with 
the  instruction  of  apprentices. 

Reflection  on  methods,  and  endeavors  to  train  novices  by 
the  experience  and  mistakes  of  older  craftsmen,  formed  a  sort 
of  groundwork  of  experience,  and  tended  to  develop  a  nomen- 

iThe  Egyptian  pyramid  of  Cochrome  is  referred  by  archaeologists  to 
the  first  dynasty  of  Manetho,  3600  B.C.,  making  it  fifty-five  centuries 
old.  It  exhibits  well  developed  skill  in  the  trades,  "dating  from  a  time 
nearly  coincident,  according  to  Biblical  authority,  with  the  creation  of  the 
world  itself  (3761  B.C.)" — Reber,  History  of  Ancient  Art,  p.  3.  See  also, 
Petrie;  Maspero;  Perrot  and  Chipiez. 

^The  Egyptians'  sculptured  wall  reliefs  and  wall  paintings  exhibit  con- 
siderable specialization  in  the  trades  several  thousand  years  B.C.  As  for 
the  Greeks,  the  picture  of  Vulcan's  smithy  in  Iliad  XVIII  is  that  of  a 
most  busineslike  and  efficient  shop.  There  is  no  mention  of  iron  or  steel, 
but  it  indicates  the  tools  employed  1000  B.C. 

So  speaking  he  withdrew,  and  went  where  they  lay  589 

The  bellows,  turned  them  toward  the  fire,  and  bade 

The  work  begin.      From  twenty  iellows  came 

Their  breath  into  the  furnaces, — a  blast 

Varied  in  strength  as  need  might  he;  .  .  . 

And  as  the  work  required.     Upon  the  fire 

He  laid  impenetrable  brass,  and  tin  595 

And  precious  gold  and  silver;  and  on  its  block 

Placed  the  huge  anvil  and  took  the  ponderous  sledge 

And  held  the  pincers  in  the  other  hand. 


When  the  great  artist  Vulcan  saw  his  task  757 

Complete,  he  lifted  all  that  armor  up 

And  laid  it  at  the  feet  of  her  who  bore 

Achilles.     Like  a  falcon  in  her  flight, 

Down  plunging  from  Olympus  capped  with  snow, 

She  bore  the  shining  armor  Vulcan  gave. 

William  Cullen  Bryant's  Translation. 


1 6  THE   SCIENCE   OF  MECHANICS. 

clature,  and  a  set  of  rules.  This  indicates  in  its  very  genesis 
the  practical  and  economical  character  of  mechanical  science. 
It  generalizes  experience.  It  is  not  only  a  mental  labor-sav- 
ing device,  but  also  a  guide  to  the  fashioning  of  physical 
labor-saving  apparatus. 

Mechanics  began,  then,  with  the  theory  and  rules  of  the 
trades.  The  very  common  origin  of  its  twin-brother  geometry, 
is  seen  on  translating  this  Greek  word  into  English :  Teco/xeTpia, 
rj, — the  science  of  measuring  the  earth.^  Herodotus  attributes 
the  origin  of  this  science  to  the  necessity  of  resurveying  the 
Egyptian  fields  after  each  inundation  of  the  Nile  and  refers 
to  the  system  of  taxation  of  Rameses  II  (c.  1340-1273  B.C.), 
which  required  such  survey.  Early  geometry  was  therefore  a 
crude  theory  of  land  surveying.  Its  abstractions  and  rules 
were  brought  to  bear  upon  mechanical  problems  and  there 
followed  that  intimate  connection  in  the  development  of  these 
sciences  which  has  been  so  useful.  Formal  mechanics  has  in- 
deed been  called  by  one  of  the  masters,^  a  geometry  of  four 
dimensions,  i.  e.,  the  three  spatial  dimensions  and  time. 

The  Ahmes  papyrus  of  the  British  Museum,  "Directions  for 
Obtaining  Knowledge  of  all  Dark  Things"  (about  2000  B.C.), 
is  perhaps  the  oldest  treatise  on  arithmetic  in  existence. 
The  Egyptians  appear  to  have  had  manuscripts  on  arithmetic 
as  early  as  2500  B.C.  But  what  every  school  boy  is  now 
taught  was  then  a  dark  mystery  known  to  but  a  few  priests 
and  scribes.  The  hieroglyphic  numerals  are  a  vertical  line 
for  I,  a  kind  of  horse  shoe  for  10,  a  spiral  for  100,  a  pointing 
finger  for  10,000,  a  frog  for  100,000  and  the  figure  of  a  man 
in  the  attitude  of  wonder  for  1,000,000;  a  rather  hopeless 
notation  for  mechanical  calculations  from  the  modern  point 
of  view. 

Building  on  the  accumulations  of  Egyptian  and  Phoenician 
civilization,  the  Greeks  began  the  Science  of  Mechanics  by 
applying  in  the  trades  the  rules  of  geometry  and  the  inductive 
and  deductive  methods  of  thought.     They  labored  under  the 

1  Pickering's  Greek  Lexicon;  Aristoph.  Nub.  202;  Th.  yia  and  tiirpov;  also 
Herodt. 

^Joseph  Louis  Lagrange  (1736-1813). 


THE   SCIENCE   OF  MECHANICS.  I7 

erroneous  conceptions  of  nature  taught  in  their  mythological 
religion,  and  they  were  further  handicapped  by  the  notion 
that  it  was  not  necessary  to  investigate  nature  at  first  hand, 
but  that  the  scheme  of  things  could  be  evolved  by  ratiocina- 
tion. 

Mechanics  as  a  science  may  be  said  to  begin  with  the  Greeks, 
as  they  formulated  the  fl^=st  principle  of  mechanics.  But  their 
speculations  were  limited  to  problems  of  equilibrium,  that  is, 
to  Statics.  They  never  evolved  any  rational  theory  of  moving 
bodies.  Dynamics  was  unknown  to  them  and  did  not  take 
form  as  a  branch  of  mechanics  until  about  1600  A.D.  The 
great  bulk  of  the  correct  theory  of  mechanics  known  in  an- 
tiquity is  commonly  attributed  to  Archimedes.  Before  con- 
sidering his  work,  it  will  be  profitable  to  glance  at  the  work 
of  several  of  his  predecessors. 

Thales,  probably  of  Greek  and  Phoenician  ancestry,  tradi- 
tion declares,  brought  the  art  of  geometry  from  Egypt  into 
Greece  about  600  B.C.  He  taught  half  a  dozen  theorems  by 
the  inductive  method.  Proclus  a  Greek  teacher  of  about  450 
A.D.,  speaks  of  him  as  the  father  of  geometry  in  Greece,  and 
declares  that  he  learned  it  in  Egypt. 

His  method  was  later  extended  by  Pythagoras  who,  about 
500  B.C.,  prepared  two  books  of  geometry  on  the  deductive 
plan.  He  appears  to  have  been  the  first  to  separate  clearly 
the  studies  of  geometry  and  of  numbers.  By  pointing  out 
that  quantity  is  incommensurable,  but  that  measure  of  quan- 
tity or  a  unit  may  be  enumerated  or  counted,  he  drew  the 
distinction  between  geometry  and  arithmetic,  and  set  apart 
the  study  of  numbers  or  arithmetic  as  a  branch  of  mathematics. 

One  finds  it  difficult  to  realize  the  mysticism  and  magic  with 
which  so  commonplace  an  idea  as  a  number  was  then  mingled. 
Pythagoras  regarded  numbers  as  having  celestial  natures,  the 
even  numbers  as  feminine  and  the  odd  as  masculine!^ 

Hippocrates  (420  B.C.)  invented  the  method  of  reducing 
one  theorem  to  another  for  proof  instead  of  going  back  to 
the  axioms  with  each  proposition;  while  Eudoxus  (355  B.C.) 
invented  proportion  and  devised  the  method  of  exhaustions, 

^"The  Philosophy  of  Arithmetic,"  Dr.  Edw.  Brooks. 


1 8  THE   SCIENCE  OF  MECHANICS. 

one  form  of  the  idea  of  limits  which  he  applied  in  geometry. 

About  300  B.C.,  Euclid  collected  and  systemized  the  geom- 
etry and  number-work  of  his  time,  invented  some  new  propo- 
sitions and  made  a  volume  on  the  "Elements  of  Geometry." 
This  work  of  fifteen  books  remained  the  standard  text-book 
of  geometry  the  "Euclid,"  of  the  following  twenty  centuries. 
The  work  gives  rules  for  the  geometrical  construction  of  various 
figures,  as  well  as  the  proof  of  numerous  theorems.  He  also 
wrote  a  volume  on  Conies  and  Geometrical  Optics. 

Aristotle  (384-322  B.C.),  the  famous  Greek  teacher,  often 
mentioned  as  one  of  the  founders  of  Science,  is  notable  for 
his  voluminous  writings  on  philosophy,  on  natural  history  and 
on  geometry,  which  in  part  directed  attention  to  the  study  of 
nature  by  direct  observation.  But  there  is  no  doubt  that  his 
teaching  on  the  theory  of  motion  and  some  of  his  notions  on 
equilibrium  were  erroneous.  His  great  reputation  as  a  natural 
philosopher  gained  acceptance  for  some  of  his  opinions  for 
eighteen  centuries  after  his  time,  and  as  they  were  wrong,  this 
was  a  great  impediment  to  the  development  of  the  science  of 
mechanics.  Even  as  late  as  1590,  Galileo  felt  the  strength  of 
the  partisans  of  the  erroneous  Aristotelian  philosophy  who 
forced  him  from  the  University  of  Pisa. 

By  200  B.C.,  four  centuries  after  Thales,  the  Greeks  had 
brought  their  geometry  to  a  high  stage  of  perfection.  Apol- 
lonius,  of  Perga  (d.  205  B.C.),  published  about  this  time  a 
treatise  on  conic  sections  and  geometry  containing  over  four 
hundred  problems  which  left  little  for  his  successors  to  improve. 
His  problem,  "to  draw  a  circle  tangent  to  three  given  circles 
in  a  plane,"  found  in  his  tretaise  on  "Tangency,"  has  baffled 
many  later  mathematicians. 

His  studies  on  astronomy  were  the  basis  of  Ptolemy's  expo- 
sition of  planetary  motions  and  his  goemetry  has  been  dis- 
covered in  two  distinct  Arabic  editions,  indicating  its  influence 
on  Moorish  mathematics  of  the  ninth  and  tenth  centuries.  He 
also  wrote  on  methods  of  arithmetical  calculation  and  on  statics, 
but  this  work  is  overshadowed  by  that  of  his  contemporary 
Archimedes,  who  appears  to  have  co-ordinated  the  scattered 
information  on  statics  and  to  have  contributed  largely  to  it. 


THE   SCIENCE   OF  MECHANICS.  I9 

What  Euclid  did  for  geometry  Archimedes  tried  to  do  for  Sta- 
tics. In  this  he  was  in  part  at  least  successful.  For  he  de- 
veloped a  body  of  correct  mechanical  doctrine  which  still  finds 
place  to-day  in  our  elementary  text  books  of  this  science. 

3.   THE  CONTRIBUTIONS  OF  ARCHIMEDES. 
(287-212  B.C.) 

Archimedes,  the  greatest  mathematician  of  antiquity,  the 
son  of  a  Greek  astronomer,  had  the  advantage  of  a  good  train- 
ing in  the  schools  of  Alexandria,  and  then  retired  to  Syracuse 
in  Sicily,  where  he  devoted  himself  to  the  study  of  mathematics 
and  mechanics. 

We  know  his  work  through  the  manuscripts  and  the  books 
which  have  come  down  to  us,  and  by  references  to  him  in  the 
classics  which  give  us  some  slight  additional  data.  Some  of 
his  writings  we  have  in  the  original  Greek,  while  others  exist 
only  in  the  Latin  or  Arabic  translation.  They  may  be  briefly 
summarized  as  follows: 

Extant  Works. 

1 .  On  the  Sphere  and  Cylinder. 

Two  books  containing  sixty  propositions  relative  to  the 
dimensions  of  cones  and  cylinders,  all  demonstrated  by 
rigorous  geometric  proof. 

2.  The  Measure  of  the  Circle. 

A  book  of  three  propositions.  Prop.  I  proves  that  the  area 
of  a  circle  is  equal  to  a  triangle  whose  base  is  equal  to 
the  circumference  and  whose  altitude  is  equal  to  the 
radius.  Prop.  II  shows  that  the  circumference  exceeds 
three  times  the  diameter  by  a  fraction  greater  than  10/70 
and  less  than  10/71.  Prop.  Ill  proves  that  a  circle  is  to 
its  circumscribing  square  nearly  as  11  to  14. 

3.  Conoids  and  Spheroids. 

A  treatise  of  40  propositions  on  the  superficial  areas  and 
volume  of  solids  generated  by  the  revolution  or  conic 
sections  about  their  axis. 

4.  On  Spirals. 

A  book  of  28  propositions  upon  the  curve  known  as  the 


20  THE    SCIENCE   OF  MECHANICS. 

spiral  of  Archimedes  which  is  traced  by  a  radius  vector 
whose  length  varies  as  the  angle  through  which  it  turns. 

5.  On  Equiponderants  and  Centers  of  Gravity. 

Two  volumes  which  are  the  foundation  of  Archimedes' 
theory  of  Mechanics.  They  deal  with  statics.  The  first 
book  contains  fifteen  propositions  and  eight  postulates. 
The  methods  of  demonstration  are  those  often  given 
to-day  for  finding  the  center  of  gravity  of — 

(a)  any  two  weights, 

(b)  any  triangle, 

(c)  any  parallelogram, 

(d)  any  trapezium. 

The  second  volume  is  devoted  to  finding  the  center  of 
gravity  of  parabolic  segments. 

6.  The  Quadrature  of  the  Parabola. 

A  book  of  24  propositions  demonstrating  the  quadrature 
of  the  parabola  by  a  process  of  summation — a  kind  of 
crude  integration. 

7.  On  Floating  Bodies. 

A  treatise  of  two  volumes  on  the  principles  of  buoyancy 
and  equilibrium  of  floating  bodies  and  of  floating  para- 
bolic conoids. 

8.  The  Sand  Reckoner,  or  Arenarius. 

A  book  of  arithmetical  numeration  which  indicates  a  method 
of  representing  very  large  numbers.  He  indicates  that 
the  number  of  grains  of  sand  required  to  fill  the  universe, 
is  less  than  10^^  It  contains  an  idea  which  might  have 
been  developed  into  a  system  of  logarithms. 

9.  A   collection   of   Lemmas, — fifteen   propositions   in    plane 

geometry. 

Archimedes  is  also  credited  with  these  lost  books,  though 
some  authorities  dispute  the  fact  that  he  ever  wrote  such 
volumes;  that  he  worked  upon  the  subjects  there  is  little 
doubt. 


the  science  of  mechanics.  21 

Lost  Works.^ 

1.  On  Polyhedra. 

2.  On  the  Principles  of  Numbers. 

3.  On  Balances  and  Levers. 

4.  On  Center  of  Gravity. 

5.  On  Optics. 

6.  On  Sphere  Making. 

7.  On  Method. 

8.  On  a  Calendar  or  Astronomical  Work. 

9.  A  Combination  of  Wheels  and  Axles. 

10.  On  the  Endless  Screw  or  Screw  of  Archimedes. 
Archimedes  is  to  be  credited  with  the  development  of  a 
theory  of  the  lever,  the  principle  of  buoyancy,  the  theory  of 
numbers  and  numeration.  He  was  the  first  to  apply  correctly 
geometry  and  arithmetic  to  mechanical  problems  of  equi- 
librium, and  he  thus  founded  the  science  of  applied  or  mixed 
mathematics.  He  founded  and  developed  the  theory  of  statics 
in  reference  both  to  rigid  solids  and  fluids,  but  he  by  no  means 
completed  it.  He  developed  no  correct  theory  of  dynamics. 
The  following  quotations  from  his  book  on  Equilibrium,  or 
the  "Center  of  Gravity  of  Plane  Figures,"  give  an  insight 
to  his  mental  attitude  and  an  idea  of  his  method  of  approaching 
problems  in  mechanics. 

Book  L 

"I  postulate  the  following: 

1.  "Equal  weights  at  equal  distances  are  in  equilibrium,  and 
equal  weights  at  unequal  distances  are  not  in  equilibrium  but 
incline  toward  the  weight  which  is  at  the  greater  distance. 

2.  "If,  when  weights  at  certain  distances  are  in  equilibrium, 
something  be  added  to  one  of  the  weights,  they  are  not  in 
equilibrium  but  incline  toward  that  weight  to  which  addition 
is  made. 

3.  "Similarly,  if  anything  be  taken  away  from  one  of  the 
weights,  they  are  not  in  equilibrium  but  incline  toward  the 
weight  from  which  nothing  was  taken. 

•Accounts  of  the  recently  discovered  "lost  works"  of  Archimedes  will 
be  found  in  the  following  periodicals:  Hermes,  vol.  42;  Bulletin  of  the  Amer- 
ican Mathematical  Society,  May,  1908;  Bibliotheca  mathematica,  vol.  7, 
p.  321. 


22  THE   SCIENCE   OF  MECHANICS. 

4.  "When  equal  and  similar  plane  figures  coincide  if  applied 
to  one  another,  their  centers  of  gravity  similarly  coincide." 

5.  "In  figures  which  are  unequal  but  similar  the  centers 
of  gravity  will  be  similarly  situated.  By  points  similarly 
situated  in  relation  to  similar  figures,  I  mean  points  such  that, 
if  straight  lines  be  drawn  from  them  to  the  equal  angles,  they 
make  equal  angles  with  the  corresponding  sides." 

6.  "If  magnitudes  at  certain  distances  be  in  equilibrium 
(other)  magnitudes  equal  to  them  will  also  be  in  equilibrium 
at  the  same  distances." 

7.  "In  any  figure  whose  perimeter  is  concave  in  (one  and) 
the  same  direction  the  center  of  gravity  must  be  within  the 
figure."     This  is  the  way  he  proves  the  equilibrium  of  the 

lever. 

^^Proposition  i." 

"Weights  which  balance  at  equal  distances  are  equal." 
"For,  if  they  are  unequal,  take  away  from  the  greater  the 

difference  between  the  two.     The  remainders  will  then  not 

balance — (Postulate  3);  which  is  absurd." 
"Therefore  the  weights  cannot  be  unequal." 

^^Proposition  2." 

"Unequal  weights  at  equal  distances  will  not  balance  but 
will  incline  toward  the  greater  weight." 

"For  take  away  from  the  greater  the  difference  between  the 
two.  The  equal  remainders  will  therefore  balance  (Postulate 
i).  Hence  if  we  add  the  difference  again  the  weights  will  not 
balance  but  will  incUne  toward  the  greater  (Postulate  2)." 

Proposition  j . 

Proves  that  weights  will  balance  at  unequal  distances,  the 
greater  weight  being  at  the  lesser  distance,  by  a  similar  kind 
of  reasoning. 

Proposition  4. 

Shows  similarly  that  two  equal  weights  have  the  center  of 
gravity  of  both  at  the  middle  point  of  the  line  joining  their 
centers  of  gravity. 


THE   SCIENCE   OF  MECHANICS. 


23 


Proposition  5. 
Proves,  if  three  equal  magnitudes  have  their  centers  of 
gravity  on  a  straight  Hne  at  equal  distances,  the  center  of 
gravity  of  the  system  will  coincide  with  that  of  the  middle 
magnitude.     He  then  proves, 

Propositions  6-7. 

Two  magnitudes,  whether  commensurable  (Prop.  6)  or  in- 
commensurable (Prop.  7)  balance  at  distances  reciprocally 
proportional  to  the  magnitudes. 

I.  Suppose  the  magnitudes  .4,  5  to  be  commensurable  and 
the  points  A,  B  to  he  their  centers  of  gravity. 

Let  DE  be  a  straight  line  so  divided  that  at  C 

A  :  B  =  DC  :  CE 

We    have    then  to  prove  that,  if  A  be  placed  at  E   and 
B  at  D,  C  is  the  center  of  gravity  of  the  two  taken  together. 


A 

N 

B 

c 

Fig.  I. 


H 


K 


Since  A  and  B  are  commensurable,  so  are  DC,  CE.  Let  N 
be  a  common  measure  of  DC,  CE.  Make  DH,  DK  each  equal 
to  CE  and  EL  (on  CE  produced)  equal  to  CD.  Then  EH  = 
CD.  Since  DH  =  CE  therefore  LH  is  bisected  at  E,  as  HK 
is  bisected  at  D. 

Thus  LH,  HK  must  each  contain  N  an  even  number  of 
times. 

Take  a  magnitude  0  such  that  0  is  contained  as  many 
times  in  ^  as  iV  is  contained  in  LH  whence 


A  :0  =  LH  :N 


24  THE   SCIENCE   OF  MECHANICS. 

But 

B  :A  =  CE:DC 
=  HK  :LH 

"Hence  B  :  0  =  HK  :  N,  or  0  is  contained  in  B  as  many- 
times  as  N  is  contained  in  HK." 

"Thus  0  is  a  common  measure  of  A,  B.  Divide  LH,  HK 
into  parts  each  equal  to  N,  and  A,  B,  into  parts  each  equal  to 
0.  The  parts  A  will  therefore  be  equal  in  number  to  those  of 
LH,  and  the  parts  of  B  equal  in  number  to  those  of  HK. 
Place  one  of  the  parts  of  A  at  the  middle  point  of  each  of  the 
parts  N  of  LH,  and  one  of  the  parts  of  B  at  the  middle  point 
of  each  of  the  parts  N  of  HK. 

"Then  the  center  of  gravity  of  the  parts  of  A  placed  at 
equal  distances  on  LH  will  be  at  E,  the  middle  point  of  LH 
(Proposition  5,  Cor.  2),  and  the  center  of  gravity  of  the  parts 
of  B  placed  at  equal  distances  along  HK  will  be  at  D  the  middle 
point  of  HK. 

"Thus  we  may  suppose  A  itself  applied  at  E,  and  B  itself 
applied  at  D." 

"But  the  system  formed  by  the  parts  0  oi  A  and  B  to- 
gether is  a  system  of  equal  magnitudes  even  in  number  and 
placed  at  equal  distances  along  LK,  and,  since  LE  =  CD  and 
EC  =  DK,  LC  =  CK  so  that  C  is  the  middle  point  of  LK. 
Therefore  C  is  the  center  of  gravity  of  the  system  ranged 
along  LK. 

"Therefore  A  acting  at  E  and  B  acting  at  D  balance  about 
the  point  C" 

The  incommensurable  case. 

"Suppose  the  magnitudes  to  be  incommensurable  and  let 
them  be  {A  ±  a)  and  B  respectively.  Let  DE  be  a  line  divided 
at  C  so  that 

(A  +  a)  :  B  =  DC  :  CE 

"Then,  if  {A  +  a)  placed  at  E  and  B  placed  at  D  do  not 
balance  about  C,  (A  +  a)  is  either  too  great  to  balance  B 
or  not  great  enough." 

"Suppose,  if  possible  that  (A  +  o-)  is  too  great  to  balance  B. 


THE   SCIENCE   OF  MECHANICS. 


25 


Take  from  (A  +  a)  a.  magnitude  smaller  than  the  deduction 
which  would  make  the  remainder  balance  B,  but  such  that 
the  remainder  A  and  the  magnitude  B  are  commensurable. 


a  :     A 


B 


Fig.  2. 

"Then,  since  A,  B  are  commensurable  and 
A  :  B  <  DC  :  CR 

A  and  B  will  not  balance  (Prop.  6)  but  D  will  be  depressed. 

"But  this  is  impossible  since  the  deduction  A  was  an  in- 
sufificient  deduction  from  (A  +  a)  to  produce  equilibrium,  so 
that  E  was  still  depressed. 

"Therefore  (A  +  a)  is  not  too  great  to  balance  B;  and 
similarly  it  may  be  proved  that  B  is  not  too  great  to  balance 
{A  +  a). 

"Hence  (A  -\-  a),  B  taken  together  have  their  center  of 
gravity  at  C." 

Thus  it  is  seen  that  the  demonstration  rests  upon  the  axiom 
that  equal  bodies  at  the  ends  of  equal  arms  of  a  rod  supported 
at  its  middle  point  will  balance  each  other.  From  this  he 
proves  that  the  bodies  will  be  in  equilibrmm  when  their  dis- 
tances from  the  fulcrum  are  inversely  as  their  weight,  and  all 
his  determinations  are  based  on  these  propositions.  All  his 
investigations  are  limited  to  -the  case  of  forces  perpendicular 
to  straight  lever  arms,  he  does  not  appear  to  have  grasped 
the  idea  of  "moments"  or  of  "equal  work"  up  and  down. 
These  conceptions  were  not  fully  attained  until  eighteen 
centuries  later. 

To  Archimedes  also  belongs  the  fame  of  establishing  the 
principle  of  buoyancy  commonly  known  as  Archimedes'  prin- 


26  THE   SCIENCE  OF  MECHANICS. 

ciple.  The  account  of  this  discovery  given  by  Vitruvius  in 
De  Architectura,  Liber  IX,  is  as  follows:  "Although  Archi- 
medes discovered  many  curious  things  proving  his  great  intelli- 
gence, that  which  I  now  narrate  is  the  most  remarkable. 
Hiero,  when  he  obtained  the  regal  power  in  Syracuse,  having 
on  the  happy  turn  of  his  fortunes  decreed  a  votive  crown  of 
gold  to  be  placed  in  a  certain  temple,  commanded  it  to  be  made 
of  great  value,  and  assigned  for  the  purpose  an  appropriate 
weight  of  metal  to  the  goldsmith.  The  latter  in  good  time 
presented  the  crown  to  the  king  beautifully  wrought  and  of 
correct  weight. 

But  a  report  having  been  circulated,  that  some  of  the  gold 
had  been  supplanted  with  silver  of  equal  weight  Hiero  was 
indignant  at  the  fraud,  and  appealed  to  Archimedes  for  a 
method  by  means  of  which  the  theft  might  be  detected. 
Charged  with  this  commission  he  by  chance  went  to  a  bath, 
and  on  getting  into  the  tub  perceived  that  just  in  proportion 
that  his  body  became  immersed,  in  the  same  proportion  the 
water  ran  out  of  the  vessel.  Whence  catching  at  the  method 
to  be  used  in  solving  the  king's  difficulty  he  leapt  out  of  the 
vessel  in  joy,  and  ran  naked  shouting  in  a  loud  voice,  evprjKa, 
I  have  found  it!" 

It  seems  that  Archimedes'  conception  was,  that  a  body 
immersed  in  water  must  raise  an  equivalent  quantity  of  water 
just  as  though  the  body  lay  on  one  arm  of  a  balance  and  the 
water  on  the  other  arm.  Buoyancy  he  conceived  as  a  case 
of  equilibrium  or  equipoise  by  a  balance  of  weights.  If  the 
object  overbalances  the  water  displaced  it  sinks.  These  ideas 
he  elaborated  in  his  book  on  Floating  Bodies. 

One  of  his  fundamental  assumptions  in  this  work  is  that 
it  is  an  essential  property  of  a  liquid  that  the  portion  that 
suffers  less  pressure  is  forced  upward  by  that  which  suffers 
greater  pressure  and  that  each  part  of  the  liquid  suffers 
pressure  from  the  portions  directly  above  it,  if  the  latter  be 
sinking  or  suffer  from  another  portion.  From  this  he  elabo- 
rates the  ideas, 

(i)  That  when  a  heavy  body  is  entirely  surrounded  by  liquid 
it  is  buoyed  up  or  balanced  in  part,  by  a  force  equal  to  the 
weight  of  the  liquid  it  displaces; 


THE   SCIENCE  OF  MECHANICS.  27 

(2)  That  when  bodies  lighter  than  a  fluid  are  wholly  im- 
mersed in  it,  they  displace  an  amount  of  liquid  greater  than 
their  own  weight  and  so  if  left  free  to  adjust  themselves  they 
rise  to  the  surface  and  float  so  that  only  so  much  of  their  bulk 
is  submerged  as  will  displace  sufficient  liquid  to  balance  them- 
selves ; 

(3)  When  a  submerged  body  displaces  a  magnitude  of  liquid 
which  just  balances  itself  it  is  in  equilibrium  anywhere  below 
the  surface  of  the  liquids. 

It  follows  from  the  story  of  the  gold  and  silver  crown  that 
Archimedes  must  have  arrived  at  the  idea  of  relative  density 
or  specific  gravity  but  he  could  not  distinguish  mass  and  weight. 
The  favorite  word  in  his  discussions  is  the  abstract  mathemati- 
cal term  magnitude  by  which  he  often  seems  to  mean  mass. 
But  how  mass  gets  that  drag  downwards,  or  how  the  force  or 
weight  is  related  to  mass  he  did  not  attempt  to  explain.  Cer- 
tainly he  presents  no  theory  on  the  subject  in  any  of  his  extant 
works. 

Some  convenient  mechanical  appliances  are  by  tradition 
commonly  attributed  to  Archimedes,  notably  the  Archimedean 
screw  or  pump,  a  device  said  to  have  been  invented  by  him 
while  in  Egypt  for  use  in  the  irrigation  works.  His  practical 
inventions  indicate  that  he,  in  common  with  all  the  eminent 
masters,  was  not  so  lost  in  his  theoretical  studies  as  to  be  out 
of  touch  with  practical  affairs. 

It  should  be  noted  that  the  geometry  of  Euclid  was  a  geom- 
etry of  forms  and  positions  whereas  that  of  Archimedes  was 
a  geometry  of  measurement.  This  new  trend  is  seen  in  the 
attention  that  Archimedes  gives  to  problems  of  the  quadrature 
of  curvilinear  plane  figures  (such  as  the  parabola),  and  to  the 
cubature  of  curved  surfaces.  This  development  of  geometry 
placed  it  in  most  intimate  connection  with  mechanics,  for 
progress  in  the  latter  depended  upon  accuracy  of  measure- 
ment. 

Therefore  we  are  indebted  to  Archimedes  not  only  for  the 
mechanical  devices  and  rules  commonly  associated  with  his 
name,  but  also  for  having  given  to  geometry  that  trend  of  de- 
velopment into  a  science  of  measurements  which  made  it  of 


28  THE   SCIENCE   OF  MECHANICS. 

such  assistance  in  developing  mechanics.  Archimedes  himself 
well  illustrated  this  in  the  field  of  statics.  Later  when  the 
labors  of  Galileo  and  Stevinus  had  developed  the  method  of 
representing  forces,  velocities  and  acceleration  by  lines,  it  was 
by  similar  geometrical  methods  that  Newton  in  his  Principia 
presented  the  proofs  of  theorems  in  dynamics. 

Ctesibius  and  Hero  (cir.  150  B.C.)  are  sometimes  mentioned 
as  the  successors  of  Archimedes.  Following  Archimedes' 
method  they  formulated  a  table  of  mechanical  appliances  set- 
ting forth  the  five  simple  principles  or  "simple  machines," 
about  as  they  are  listed  in  our  elementary  textbooks  of  physics 
to-day.  But  though  they  made  several  practical  inventions 
such  as  the  forcing  pump,  the  clepsydra  and  air-gun  and  con- 
trived curious  fountains  and  syphons,  they  do  not  appear  to 
have  added  anything  to  the  principles  of  mechanics,  nor  do 
they  appear  to  have  comprehended  the  theory  of  their  mechan- 
ical appliances,  except  in  so  far  as  the  principles  of  Archimedes 
could  explain  them. 

There  is  no  evidence  to  show  that  the  principle  of  work 
was  understood  or  appreciated  in  ancient  times  in  spite  of 


Fig.  3. 

the  fact  that  we  feel  almost  instinctively  now,  that  in  a  lever 
such  as  Fig.  ;t,  the  force  times  the  distance  it  moves  {i.  e., 
the  work  applied),  is  equal  to  the  resistance  times  the  distance 
it  moves,  {i.  e.,  the  work  done)  if  we  ignore  friction,  and  that 
the  algebraic  sum  of  the  positive  and  negative  work  is  zero. 
The  simple  equation  of  work,  F  X  S  =  R  X  S'  was  Chinese  to 
Archimedes,  for  algebraic  symbolic  notation  was  not  known 
in  mechanics  in  his  time.     Archimedes  does  not  appear  to  have 


THE   SCIENCE   OF  MECHANICS.  29 

attained  to  the  conception  of  "moment,"  nor  "principle  of 
work,"  nor  "conservation  of  center  of  gravity."  He  made 
equilibrium  in  the  lever  depend  on  the  length  of  the  lever-arm 
and  the  "magnitude"  of  the  bodies  hung  on  the  ends  of  the 
lever-arms  without  understanding  the  terms  moment,  mass, 
work  or  weight  in  the  modern  sense. 

The  physical  science  of  "the  Greeks  was  limited  to  calcula- 
tions based  upon: 

(i)   The  law  of  the  lever, 

(2)  Center  of  gravity, 

(3)  Density, 

(4)  Hydrostatic  pressure, 

(5)  Arithmetical  relations  of  tones, 

(6)  The  law  of  the  reflection  of  light. 
Ancient  Greece  was  a  slave  country.     At  the  height  of  its 

glory  Athens  contained  twenty  slaves  to  one  free  citizen.  The 
slaves  were  permitted  no  initiative  and  there  was  no  incentive 
to  mechanical  invention.  Indeed,  the  application  of  natural 
forces  and  the  substitution  of  machines  for  slave-labor  would 
have  been  viewed  with  alarm  by  all  classes  of  the  Greek  state 
as  ushering  in  an  industrial  and  social  revolution.  Inventors 
and  innovators  therefore  met  scant  encouragement  in  ancient 
Greece. 

The  government  was  a  close  corporation  of  capitalist  citizens 
whose  profits  depended  upon  the  slaves.  Furthermore  it  was 
to  the  interest  of  the  government  to  keep  the  slaves  steadily 
employed  yet  not  oppressively  burdened.  Conditions  of  life 
were  favorable  in  the  Greek  peninsula,  and  history  records 
very  few  slave  insurrections.  There  was  no  urgent  demand  for 
mechanical  invention  and  no  reward  for  it.  Only  free  men 
have  an  interest  in  the  improvement  of  their  tools  and  only 
under  the  laws  of  property  and  of  patents  is  there  encourage- 
ment and  incentive  to  mechanical  invention. 

With  the  extension  of  the  Roman  power  on  the  fall  of 
Syracuse,  in  which  Archimedes  lost  his  life,  conditions  were 
not  favorable  to  the  advance  of  science.  The  Romans,  a 
practical,  commerical,  military  people  did  not  advance  the 
theory  of   mechanics.     Their   talents   lay   in  administration 


30  THE   SCIENCE   OF  MECHANICS. 

rather  than  in  science.  They  used  the  simple  machines  prac- 
tically and  successfully  on  land  and  on  sea,  in  war  and  in 
peace,  and  by  trireme  and  catapult  extended  their  dominon 
over  the  known  world.  In  their  marvellous  public  works 
aqueducts,  baths,  fountains,  sewers,  roads,  public  buildings, 
and  monuments,  we  find  examples  of  the  art  of  construction 
rather  than  of  the  science  of  engineering. 

They  built  by  "rule  of  thumb"  and  experience  based  on 
trial,  using  a  large  surplus  of  materials.     No  delicate  appre- 
ciation of  stresses  is  apparent  in  their  architecture.     It  is 
massive,  heavy  and  monumental,  without  subtlety  of  artistic 
conception  or  of  scientific  design.     It  is  true  the  Romans  in- 
troduced as  common  features  in  their  buildings,  arches,  vaults 
and  domes  which  were  used  by  the  Greeks,  Persians  and  Egyp- 
tians   but    rarely,  but  they  used    them  without  theoretical 
calculation.     Some  of  their  buildings  were  supplied  with  run- 
ning water  carried  in  lead  pipes,  and  were  heated  by  hot  air 
in  tile  flues  but  they  had  not  grasped  even  the  elements  of 
hydraulics  or  thermodynamics.     It  was  not  till  after  1600  A.D.  / 
that  the  principle  of  moments  and  the  law  of  action  and  re-  ^ 
action  upon  which  the  common  engineering  calculations  are^ 
based,  were  fully  apprehended. 

Nor  did  Roman  philosophers  and  writers  busy  themselves 
with  mechanical  science.  The  works  of  Lucretius  (95-52 
B.C.),  Vitruvius  (85-26  B.C.),  Seneca  (2-66  A.D.)  and  Pliny 
(23-79  A.D.)  contain  no  new  idea  in  mechanics.  For  prac- 
tically twenty  centuries  no  advance  was  made  in  the  theory 
of  mechanics  after  the  time  of  Archimedes.  "Vir  stupendae 
sagacitatis,  qui  prima  fundamenta  posuit  inventionum  fere 
omnium  in  quibus  promovendis  setas  nostra  gloriatur"  is  the 
tribute  Wallis  penned  two  thousand  years  later,  when  Latin 
was  still  the  language  of  scholars  and  engineers. 

The  Romans  left  their  mark  on  civilization  as  the  annals  of 
government,  law  and  language  testify,  but  there  does  not 
appear  to  their  credit  the  discovery  of  a  single  scientific  prin- 
ciple or  the  invention  of  an  important  mechanical  appliance 
for  mitigating  the  drudgery  and  toil  of  mankind.  Their  slaves 
and  captives  labored  long  and  hard,  tilling  the  fields,  in  the 
galleys,  or  with  brick,  tiles  and  concrete  on  the  aqueducts. 


THE   SCIENCE   OF  MECHANICS.  3I 

Every  conquest  delivered  to  the  Imperial  City  a  new  supply 
of  labor.  Besides,  in  tranquil  times  the  legions,  kept  from  mis- 
chief by  employment  on  roads,  bridges  and  wall  building, 
supplied  abundant  labor.  In  a  word  the  Romans  had  neither 
interest  in  the  theory  of  mechanics,  nor  the  pressing  necessity 
for  improved  mechanical  appliances,  as  they  commanded  an 
abundance  of  cheap  labor.  For  these  reasons  this  intensely 
practical  people  appears  to  have  made  no  contribution  to  the 
science  of  mechanics. 


32  THE   SCIENCE  OF  MECHANICS. 


REFERENCES. 

Ball,  J.  J.     The  History  of  Mathematics. 

Burr,  W.  H.     Ancient  and  Modern  Engineering. 

Darwin,  C.     Descent,  Origin  of  Species,  etc. 

Diihring.     Geschichte  der  Principien  der  Mechanik. 

Faraday,  M.     Proc.  Royal  Ins.  G.  B. 

Fletcher,  B.     History  of  Architecture. 

Grote.     History  of  Greece. 

Hamlin,  A.  D.  F.     History  of  Architecture. 

Heiburg,  Leipsic  (1881);  Heath,  Cambridge  (1891).     Opera  Archimedis 

Huxley,  T.  H.     Science  and  Educ,  Man's  Place  in  Nature. 

Lubbock,  J.  J.     Prehistoric  Times. 

Mach,  E.     The  Science  of  Mechanics. 

Maspero  and  Sayce.     The  Dawn  of  Civilization. 

Perrot  and  Chipiez.     History  of  Ancient  Art. 

Petrie,  W.  M.  F.     A  History  of  Egypt;  Tales;  Royal  Tombs. 

Reber,  F.     History  of  Ancient  Art. 

Renan,  E.     Mission  de  Phenicie. 

Robinson,  J.  H.     A  History  of  Western  Europe. 

Rogers,  New  York  (1900).     History  of  Babylonia  and  Assyria. 

Spencer,  H.     First  Principles,  Education. 

Stark.     Archaologie  der  Kunst. 

Tylor.     Primitive  Culture. 

Tylor.     The  Early  History  of  Mankind. 

Tyndall,  J.     Essays,  Notes  and  Papers. 

Winckelmann.     Geschichte  der  Kunst  des  Alterthums. 

Winckler  (1900).     Die  politische  Entwicklung  Babyloniens  und  Assyriens. 

Wright,  D.  F.     Man  and  the  Glacial  Period,  etc. 


PART   II. 
I.  THE  MEDIAEVAL  PERIOD,   500-1500  A.D. 

I.  The  Medieval  Attitude  toward  Science. 

The  period  of  societal  reconstruction  which  followed  the 
decay  of  the  Roman  Empire  was  not  a  time  of  scientific  re- 
search or  achievement.  It  was  an  age  of  semi-barbarism, 
tumult  and  superstititon.  Those  of  gentle  and  scholarly  dis- 
position who  sought  the  quiet  asylum  of  the  Church  found 
there  a  faith  in  an  established  cosmography,  which  did  not 
encourage  independent  research  and  investigation  of  natural 
phenomena. 

Dr.  Andrew  D,  White,  of  Cornell,  says:^  "The  establishment 
of  Christianity,  beginning  a  new  evolution  of  theology,  arrested 
the  normal  development  of  physical  sciences  for  fifteen  hundred 
years."  This  is  in  part  true  and  it  was  due,  during  the  first 
thousand  years  at  least,  to  a  widespread  belief,  based  on  the 
New  Testament,  that  the  end  of  the  world  was  soon  at  hand. 
St.  Paul  had  preached:  "For  ye  know  perfectly  that  the  day 
of  the  Lord  so  cometh  as  a  thief  in  the  night,"  and  St.  Peter 
had  reiterated:  "The  day  of  the  Lord  will  come  as  a  thief  in 
the  night  in  the  which  the  heavens  shall  pass  away  with  a 
great  noise  and  the  elements  shall  melt  with  fervent  heat  and 
the  earth  also  and  the  works  that  are  therein  shall  be  burned 
up." 

It  was  widely  proclaimed  that  the  world  was  in  its  last  days, 
that  just  as  the  antediluvian  world  was  destroyed  in  the  flood, 
so  now  the  coming  of  the  Lord  in  a  cataclysm  of  fire  was  to  be 
awaited  from  day  to  day.  With  such  a  stupendous  supernatural 
event  impending  and  the  termination  of  the  world  imminent, 
devotion  to  mechanical  science  was  sheer  folly.  Even  such 
science  as  had  been  developed  was  now  become  vain  and  trivial, 
and  was  neglected  in  the  face  of  the  duty  to  watch  and  to  pray. 

The  end  of  the  world  was  announced  for  various  specific 

^"Warfare  of  Science  and  Theology,"  vol.  i,  p.  375. 
3  33 


34  THE   SCIENCE  OF  MECHANICS. 

dates  notably  looo  A.D.,  and  all  endeavor  except  "saving 
souls"  was  pronounced  folly  and  the  inspiration  of  the  evil 
one.  And  when,  after  centuries  of  waiting,  the  existing  order 
was  found  going  along  just  as  ever,  and  curious  men  began 
to  turn  again  to  worldly  affairs,  they  found  theology  had  woven 
a  magic  circle  and  defied  any  one  to  find  truth  outside  of  it. 
In  place  of  verified  experience,  a  literal  belief  in  the  Old 
and  New  Testament  offered  a  precarious  theophany  and 
created  a  frenzied  terror  of  supernatural  agencies.  Demons, 
(  imps  and  devils  rode  the  wind  and  disported  themselves  to 
the  fevered  imagination  of  the  time  as  the  cause  of  the  most 
common  occurrences.^  Any  prying  into  the  secrets  of  nature 
was  held  to  be  dangerous  to  body  and  soul.  Physics  and 
chemistry,  such  as  there  was,  were  tabooed  as  the  devil's 
own  arts,  and  experimental  research  was  anathema. 
I  Stories  of  interference  with  the  law  of  gravitation  by  the 
\  devil  and  the  saints  are  common  among  the  legends  of  this 
\period.  A  story  published  in  the  Dialogues  of  St.  Gregory 
the  Great,  Vol.  II,  illustrates  this  belief.  During  the  con- 
struction of  Monte  Cassino  about  530,  one  day  the  builders 
found  a  stone  which  their  united  efforts  could  not  move.  They 
reported  this  to  St.  Benedict,  "who  instantly  knew  the  devil 
was  hanging  on  to  it."  He  exorcised  the  devil  and  the  stone 
which  before  was  too  heavy  for  six  men  became  so  light  that  St. 
Benedict  lifted  it  with  ease  and  put  it  into  the  wall.  A  similar 
account  of  the  devil  increasing  the  gravity  of  two  marble 
columns  at  the  Cathedral  of  St.  Virgile,  Bishop  of  Aries,  about 
600,  is  given  in  "Les  Petits  Bollandistes,"  Vol.  Ill,  p.  162. 

Even  after  the  year  1000  A.D.,  ideas,  which  to  us  appear 
most  fantastic,  were  handed  down  for  generations  apparently 
without  anyone  doubting  their  verity  or  making  any  endeavor 

ipor  the  spirit  of  the  time  refer  to  Longfellow's  "Christus;  a  mystery." 

Safe  in  this  Wartburg  tower  I  stand 

Where  God  hath  led  me  by  the  hand,  .  .  . 
Safe  from  the  overwhelming  blast 
Of  the  mouths  of  Hell,  that  followed  me  fast, 

And  the  howling  demons  of  despair 

That  hunted  me  as  a  beast  to  his  lair. 
(Second  interlude.) 


THE   MEDIEVAL   PERIOD.  35 

to  verify  them  by  experiment.  When  Albertus  Magnus  (cir. 
1250),  a  famous  philosopher  of  the  thirteenth  century,  beHeved 
that  the  diamond  could  be  softened  in  the  blood  of  a  stag  fed 
on  parsley  and  that  a  sapphire  would  drive  away  boils,  it  is 
hardly  to  be  expected  that  even  the  learned  of  this  period 
would  have  any  conception  or  appreciation  of  a  science  of 
mechanics. 

Though  St.  Paul  had  advised,  "Prove  all  things,  hold  fast 
to  what  is  good,"  St.  Augustine  commanded  in  vigorous  Latin 
— "Major  est  Scripturse  auctoritas  quamomnis  humani  ingenii 
capacitas,"  i.  e.,  accept  nothing  except  on  authority  of  Scrip- 
ture for  that  is  greater  than  all  the  powers  of  the  human  mind. 
When  asked,  might  there  not  be  inhabitants  on  the  other  side 
of  the  earth,  he  answered,  it  is  impossible  that  there  should 
be  inhabitants  on  the  other  side  of  the  earth,  for  on  judgment 
day  such  men  could  not  see  the  Lord  descending  through  the 
air.  Discussion  was  closed  by  authority  and  debate  came 
to  be  restricted  to  such  questions  as,  whether  an  angel  in 
passing  from  one  spot  to  another,  had  to  pass  through  the 
intervening  space. 

It  came  to  be  considered  blasphemous  to  wish  for  or  to 
attempt  to  better  earthly  conditions,  and  presumptuous  to 
attempt  to  explain  phenomena  except  in  terms  of  mystic 
theology.  So,  in  the  course  of  the  centuries,  an  unfortunate  1 
conviction  was  developed  that  science  was  dangerous  and  evil.  \ 
This  persisted  beyond  the  Reformation.  Martin  Luther  (1483- 
1 546)  complained : '  'The  people  give  ear  to  an  upstart  astrologer 
(Copernicus)  who  strives  to  show  that  the  earth  revolves, — 
but  Sacred  Scripture  tells  us  that  Joshua  commanded  the 
sun  to  stand  still,  not  the  earth." 

In  much  the  same  spirit,  Melanchthon  (1497-1560)  declared: 
"It  is  the  want  of  honesty  and  decency  to  say  that  the  earth 
revolves  and  the  example  is  pernicious.  It  is  the  part  of  a  good 
mind  to  accept  the  truth  as  revealed  by  God  and  to  acquiesce 
in  it."  Indeed,  theologians  of  all  persuasions,  have,  at  some 
time,  denounced  the  Copernican  idea,  for  Scripture  declares  the 
"sun  Cometh  forth  as  a  bridegroom" — and  "the  earth  standeth 
fast  forever."     When  a  theologian  did  deign  to  debate  such  a 


36  THE  SCIENCE  OF  MECHANICS. 

topic,  his  argument  was  likely  to  be  like  that  of  Fromundus  of 
Antwerp  who,  in  refuting  the  revolution  of  the  earth,  declared 
that  "the  buildings  would  fly  off  with  such  rapid  motion,  and 
that  men  would  have  to  be  provided  with  claws  like  cats  to 
enable  them  to  hold  onto  the  earth's  surface." 

The  theologian  who  declared  in  Galileo's  time  (1600)  that 
"geometry  is  of  the  devil,"  and  "that  mathematicians  should 
be  banished  as  the  authors  of  all  heresies,"  was  but  fanatically 
defending  his  traditions.  The  matter  is  summed  up  by  Huxley 
in  his  Essay  on  Science  and  Culture  (p.  145),  where  he  says: 
"The  business  of  the  philosopher  of  the  middle  ages  was  to 
deduce  from  the  data  furnished  by  theologians,  conclusions  in 
accordance  with  ecclesiastical  decrees.  They  were  allowed  the 
high  privilege  of  showing  by  logical  process  how  and  why  that 
which  the  Church  said  was  true  and  must  be  true  and  if  their 
demonstrations  fell  short  of  or  exceeded  this  limit,  the  church 
was  maternally  ready  to  check  their  aberrations;  if  need  be  by 
the  secular  arm. 

Between  the  two,  our  ancestors  were  furnished  with  a  com- 
pact and  complete  criticism  of  life.  They  were  told  how  the 
world  began  and  how  it  would  end ;  they  learned  that  material 
existence  was  a  base  and  insignificant  blot  on  the  fair  face  of 
the  spiritual  world  and  that  nature  was  to  all  intents  and 
purposes,  the  playground  of  the  devil;  they  learned  that  the 
earth  is  the  center  of  the  visible  universe,  and  that  man  is  the 
cynosure  of  things  terrestrial,  and  more  especially  was  it  incul- 
cated that  the  course  of  nature  had  no  fixed  order  but  that  it 
could  be  and  constantly  was,  altered  by  the  agency  of  innumer- 
able spiritual  beings,  good  and  bad,  according  as  they  were 
moved  by  the  deeds  and  prayers  of  man.  The  sum  and  sub- 
stance of  the  whole  doctrine  was  to  produce  the  conviction  that 
the  only  thing  really  worth  knowing  in  this  world  was  how  to 
secure  that  place  in  a  better,  which  under  certain  conditions 
the  church  promised."  There  was  no  place  in  such  a  scheme 
for  a  science  of  mechanics. 

To  the  unbiased  student  there  is  a  measure  of  truth  in  the 
remarks  of  Dr.  White  and  Dr.  Huxley  and  yet  in  justice  it 
must  be  said  that  they  also  carry  a  sting  and  a  reproach  which 


THE   MEDIEVAL   PERIOD.  37 

is  somewhat  unfair.  The  attitude  of  the  Middle  Ages  was  a 
growth;  it  was  developed  in,  and  was  not  imposed  on  Europe. 
Whatever  reproach  there  is,  should  be  placed  where  it  belongs 
on  the  general  ignorance  and  stupidity  of  the  inhabitants  of 
Europe  during  this  period.  There  was  no  conscious  conspiracy 
to  retard  progress  if  we  except  the  bigotry,  fanaticism  and 
perversion  which  inevitably  .accompany  ignorance  anywhere 
and  in  any  time.  As  the  general  average  of  intelligence  rose, 
the  situation  improved.  It  cannot  be  denied  that  the  Church 
was  the  great  conserving  and  civilizing  agency  of  this  era. 

All  the  learning  of  the  ancients  was  not  lost  or  destroyed 
outright,  but  continued  to  filter  through  Europe  until  it  gained 
force  under  the  favorable  circumstances  of  the  so-called 
Renaissance  about  1600.  But  it  is  most  unfortunate  that 
early  Christian  fanaticism  burned  and  destroyed  so  much  that 
was  good,  though  "Pagan."  In  the  East,  the  Greek  School 
at  Alexandria  preserved,  for  some  centuries  A.D.,  the  learning 
of  the  ancients,  but,  about  the  fifth  century,  there  developed 
within  the  church  a  pronounced  spirit  of  hostility  toward  the 
scientific  spirit. 

Persecutions  became  common  and  culminated  in  415  A.D. 
with  the  murder  of  Hypatia  and  the  breaking  up  of  the  Alex- 
andrian University.  In  a  burst  of  religious  fervor,  the  schools 
and  a  portion  of  the  library  were  destroyed.  The  scholars 
were  forced  to  flee  to  Byzantium,  where  schools  grew  up. 

In  the  west,  after  500  A.D.,  Roman  civilization  finally  went 
down  under  the  successive  inroads  of  barbarians  from  the 
North  and  culture  and  refinement  were  eclipsed  in  Italy. 

Under  Constantine  (306-337),  Christianity  was  recognized 
(313)  and  Byzantium  rebuilt  as  the  Eastern  capital.  Until 
476,  there  were  two  capitals  but  the  center  of  wealth  and 
population  shifted  steadily  to  this  new  "City  of  Constantine." 
Constantinople  soon  became  the  wealthiest  and  most  enlight- 
ened city  of  the  world,  a  quiet  retreat  for  scholarly  pursuits. 
Here  many  ancient  manuscripts  on  mathematics  and  mechanics 
were  read,  copied,  and  preserved  for  posterity. 

On  the  capture  of  Constantinople,  a  thousand  years  later 
in  1453,  this  Greek  and  Byzantine  learning  was  spread  to 


( 


38  THE   SCIENCE   OF  MECHANICS. 

various  cities  of  Western  Europe  by  traveling  scholars,  and 
stimulated  scholarship  in  the  West.  As  the  mediaeval  uni- 
versities grew  up  from  the  church  schools,  their  enthusiasm 
was  naturally  not  in  the  direction  of  scientific  or  mechanical 
investigation.  An  age  of  faith  is  not  inclined  to  be  an  age 
of  investigation,  and  the  mediaeval  period  developed  little 
mechanical  progress. 

The  one  notable  indirect  contribution  to  mechanics  in  this 
period  was  the  introduction  about  1200,  of  Hindu  arithmetic 
and  Arabic  algebra  into  Europe  through  the  Moors  of  Spain. 
Among  the  ancients,  primitive  number  pictures  such  as  the 
Egyptian  hieroglyphics  and  the  Babylonian  cuneiform  sym- 
bols were  used  for  the  digits. 

The  Greeks  used  the  letters  of  their  alphabet  a,  /3,  7,  8,  etc., 
to  represent  numbers.  The  Roman  system  was  little  better 
and  no  extensive  calculation  could  be  performed  without  the 
aid  of  a  registering  instrument  of  colored  beads  called  an 
abacus.  Our  present  powerful  system  of  ten  symbols,  the 
"ten  digits,"  and  the  "method  of  position,"  whereby  their 
value  depends  on  their  place,  seems  to  have  originated  with 
the  Hindus,  and  was  carried  into  Europe  by  the  Arabs. 

Leonardo  Fibonacci  of  Pisa  (1175)  among  others  is  credited 
with  introducing  it  into  Italy  by  his  book  Liber  Abaci  (1202). 
His  introduction  reads — "The  nine  figures  of  the  Hindoos  are 
9,  8,  7,  6,  5,  4,  3,  2,  I.  With  these  nine  and  with  the  sign  o 
which  in  Arabic  is  called  sifr,  any  number  may  be  written." 
It  is  likely  that  convenience  and  serviceableness  in  commerce 
brought  the  system  into  vogue  through  the  trade  of  Genoa 
and  Venice  with  the  Orient.  From  these  ports  the  merchants 
probably  spread  it  by  the  great  overland  trade  routes  through 
Nuremberg  and  the  Rhine  to  Antwerp,  Bruges  and  the 
towns  of  the  Hanseatic  League.  The  money  exchanges  and 
the  channels  of  trade  probably  had  more  to  do  with  spreading 
it  over  Europe  than  the  philosophers. 

The  college  accounts  in  the  English  Universities  are  found  to 
have  been  kept  in  Roman  numerals  up  to  about  the  year  1550 
and  even  later.  After  this  date  the  Arabic  system  generally 
displaced  the  Roman  method. 


the  medieval  period.  39 

2.  The  Influence  of  Moorish  Culture. 

Before  the  time  of  Mohammed  (570-632),  the  Arabs  had 
played  an  inactive  part  in  history;  but,  when  the  wandering 
tribes  of  the  desert  had  been  welded  into  a  nation  by  the 
fiery  enthusiasm  of  the  prophet  and  his  fanatical  followers, 
they  began  to  be  a  factor  in  the  civilization  of  both  East  and 
West. 

Within  a  decade  after  Mohammed's  death,  the  faith  had 
conquered  Arabia,  Palestine,  Syria  and  Persia,  and  within  a 
few  years  more,  the  Moslems  threatened  Europe  from  the 
northern  coast  of  Africa.  By  711  all  Spain,  except  Asturia, 
was  subject  to  their  sway,  and  their  dominion  began  to  ap- 
proach in  extent  the  glorious  Empire  of  Rome.  In  their 
opinion,  the  Koran,  the  new  revelation,  was  destined  to  sup- 
plant the  Bible. 

The  Koran  is  evidently  based  on  the  Hebrew  and  Christian 
scriptures.  Islam  is  an  offshoot  of  Christianity,  modified  to 
suit  the  Arabic  temperament,  by  a  coloring  of  Oriental  imagery 
and  fatalism.  The  theological  structure  of  both  is  the  same. 
There  is  much  the  same  scheme  of  rewards  and  punishments 
with  a  tinge  of  predestination.  The  prohibition  in  the  Koran 
against  "graven  images"  was  held  to  forbid  the  representation 
of  any  human  or  animal  form. 

This  had  a  marked  effect  upon  their  arts,  and  no  doubt  encour- 
aged the  study  of  geometry  and  mathematics  in  general.  On 
the  whole  the  Moslems  seem  to  have  been  rather  favorably  dis- 
posed toward  pagan  culture,  regarding  it  with  placid  superi- 
ority rather  than  enmity,  and  they  never  were  hostile  to  the 
scientific  spirit.  When  in  Europe  the  practice  of  medicine 
was  looked  at  askance,  the  Arabs  were  adept  in  medicine,  and 
their  surgeons  were  in  demand  in  the  courts  of  Europe. 

The  nomadic  Arabs  had  neither  need  nor  desire  for  a  science 
of  mechanics  and  the  earlier  caliphs  were  too  busy  establishing 
their  empire,  to  develop  any  of  the  arts  and  sciences.  But 
toward  the  end  of  the  eighth  century,  when  their  religious 
fervor  was  no  longer  at  a  white  heat,  the  caliphs  became 
patrons  of  learning. 


40  THE   SCIENCE  OF   MECHANICS. 

Through  the  Greeks  of  the  conquered  provinces,  the  Moham- 
medans became  acquainted  with  classical  learning  and  blended 
it  with  the  wisdom  of  the  Orient,  which  they  had  from  India 
and  Persia.  From  the  tenth  to  the  thirteenth  centuries,  the 
Arabs  were  the  teachers  of  Europe.  They  brought  about, 
within  their  dominions,  a  renaissance  of  the  Greek  culture. 

As  early  as  800,  under  the  caliphate  of  Haroun-al-Raschid, 
Bagdad  was  a  famous  center  of  culture.  Later,  the  Western 
caliphs  developed,  at  Cordova  and  Seville,  schools  and  libraries 
which  equalled  those  of  Constantinople  and  Bagdad,  and  made 
these  Spanish  cities  famous  seats  of  learning. 

In  the  tenth  century,  Cordova  was  one  of  the  greatest 
centers  of  commerce  of  the  world  and  supported  eighty  schools. 
The  University  of  Cordova,  with  its  library  of  500,000  volumes, 
became  famous  throughout  Christendom.  Philosophy,  mathe- 
matics, medicine,  geography,  astronomy  and  mechanics  were 
taught  from  Arabian  translations  of  the  masters  of  ancient 
Greece,  Persia,  and  India. 

In  working  over  this  material,  the  Moorish  scholars,  as  was 
to  be  expected,  developed  new  ideas  and  methods,  especially 
in  mathematics,  astronomy  and  alchemy.  In  mechanics  and 
geometry  they  studied  and  preserved  for  posterity  the  writings 
of  Archimedes,  Euclid,  and  Aristotle.  They  developed  known 
principles  and  perfected  methods,  but  it  does  not  appear  that 
they  made  any  very  important  advance.  Extensive  fortifica- 
tions and  irrigation  works  were  developed  by  their  engineers, 
who  were  well  versed  in  algebra  and  statics,  but  had  but  little 
grasp  of  dynamics. 

The  English  champion  of  Science,  Professor  Huxley,  may 
be  again  quoted  to  advantage  on  this  topic.  He  says,  "Even 
earlier  than  the  thirteenth  century,  the  development  of 
Moorish  civilization  in  Spain  and  the  great  movement  of  the 
Crusades  had  introduced  the  leaven  which,  from  that  day  to 
this,  has  never  ceased  to  work.  At  first,  through  the  inter- 
mediation of  Arabic  translations,  afterwards  by  the  study  of 
the  originals,  the  western  nations  of  Europe  became  acquainted 
with  the  writings  of  the  ancient  philosophers  and  poets,  and 
in  time  with  the  whole  of  the  vast  literature  of  antiquity. 


THE   MEDIiEVAL   PERIOD.  4 1 

Whatever  there  was  of  high  intellectual  aspiration  or  dominant 
capacity  in  Italy,  France,  Germany,  and  England,  spent  itself 
for  centuries  in  taking  possession  of  the  rich  inheritance  left 
by  the  dead  civilizations  of  Greece  and  Rome.  .  .  .  There 
was  no  physical  science  but  that  which  Greece  had  created." 
This  found  its  way  into  Europe  in  part  through  Arabic  trans- 
lations of  Greek  and  Latin  texts. 

The  celebrated  Moslem  scholar,  Al-Khuwarizmi  or  Mo- 
hammed Ibn  Musa  (cir.  900  A.D.)  wrote  voluminously  on 
mathematics,  on  Hindu  arithmetic,  the  sun-dial,  and  the 
astrolabe.  His  "al-jabr-w'al-muqabalah,"  that  is  the  "red- 
integration and  the  comparison,"  a  treatise  on  algebra,  gave 
the  name  to  this  science. 

The  Arabic  numbers  and  the  algebraic  method  were  an 
immense  advance  over  the  clumsy  Roman  numbers.  Without 
these,  it  is  hardly  possible  to  apply  mathematics  extensively 
to  mechanical  problems.  This  indicates  one  reason  why  the 
ancients  did  not  advance  further  in  the  practical  applications 
of  mechanical  science.  Much  advance  in  mechanics  was 
simply  impossible  with  the  old  Roman  arithmetic  which 
possessed  a  most  awkward  duodecimal  system  of  fractions. 
Decimal  fractions  date  from  1600  when  Stevinus,  the  Flemish 
engineer,  recommended  them  in  his  writings. 

Of  the  numerous  Christian  scholars  who  attended  the  Moor- 
ish Universities  of  Cordova  and  Seville,  the  most  famous  was 
Gerbert,  who  later  became  Archbishop  of  Rheims,  and  who  as 
Pope  Sylvester  H  (999-1003),  exercised  a  wide  influence  in 
Christendom.  He  is  credited  with  the  introduction  in  Europe 
of  the  Arabic  mathematics. 

The  Moors  made  small  original  contributions  to  the  science 
of  mechanics,  but  they  are  to  be  credited  with  the  preservation 
and  development  of  the  Greek  and  Indian  knowledge  of  arith- 
metic, geometry,  and  mechanics  and  the  diffusion  of  it  through- 
out Europe.  The  Arabic  words  in  our  language  indicate  the 
breadth  of  their  influence: — algebra,  alcohol,  Aldebaran, 
almanac,  amalgam,  alkali,  borax,  cipher,  carat,  minaret, 
nadir,  Vega,  zenith,  zero.  Their  invention  of  algebra  and 
development  of  the  Arabic  numerals  and  notation,  while  not 


42  THE   SCIENCE  OF  MECHANICS. 

a  contribution  to  mechanics  proper,  had  a  most  direct  bearing 
upon  the  future  progress  of  the  science,  for  without  it,  the 
development  of  our  analytical  mechanics  would  probably  have 
been  long  delayed. 

With  the  coming  of  the  ignorant  and  fanatical  Turks  under 
Genghis  Khan  in  the  middle  of  the  thirteenth  century,  Arabic 
civilization  rapidly  declined  and  the  development  of  mathe- 
matics and  mechanics  was  arrested  in  the  Moslem  domain. 
Four  hundred  years  of  the  Turks  has  made  the  once  world- 
renowned  Byzantium,  one  of  the  most  backward  cities  on  the 
globe.  It  was  not  till  about  1890  that  the  Sultan  would  permit 
a  railway  to  run  into  Constantinople. 

3.  The  Period  of  the  Renaissance. 

The  new  order  which  slowly  overcame  and  displaced  the 
conceptions  of  mediaeval  times  was  the  expression  of  a  revo- 
lution in  the  realm  of  thought.  The  Renaissance  was  a  period 
of  breaking  away  from  the  ideas  and  ideals  of  the  Middle  Ages. 
It  was  in  part  the  result  of  the  recognition  of  certain  provinces 
of  thought  and  endeavor,  which  the  mediaeval  spirit  either 
ignored  or  condemned  and  in  part  the  victory  of  certain 
superior  features  of  the  civilization  of  Athens  and  Rome.  The 
inventions  of  printing,  of  gunpowder,  of  the  mariner's  compass 
and  the  discovery  of  America,  accelerated  this  tendency,  and 
the  religious,  political  and  social  changes  followed. 

With  the  weakening  of  the  dictates  of  established  authority, 
men  credited  personal  experience  more.  They  slowly  became 
less  biased  and  more  open-minded  in  their  opinions.  Pagan 
writings,  which,  in  mediaeval  times,  were  regarded  with  aver- 
sion, if  not  fear  and  distrust,  came  to  be  studied  with  interest. 
Good  was  found  in  the  manuscripts  of  the  infidel  Moslems; 
their  writings  were  read  with  interest  and  appreciation,  and 
their  arithmetic  was  adopted  throughout  Europe.  All  this 
prepared  the  way  for  a  new  start  in  Science. 

Even  the  theologians  began  to  be  dissatisfied  with  barren 
dogmas.  One  of  the  first  to  break  with  the  prevailing  scholas- 
ticism was  Cardinal  Nicholas  of  Cusa  (d.  1464),  who  possessed 
the  independence  to  say  that  man  was  prone  to  err,  that  it 


THE   MEDIEVAL   PERIOD.  43 

was  good  to  hold  one's  opinions  lightly,  and  to  reject  them 
when  they  began  to  appear  erroneous.  He  cultivated  mathe- 
matics and  is  said  to  have  taught  an  imperfect  heliocentric 
theory. 

The  dawn  of  the  period  of  the  Renaissance  may  be  set 
at  about  1450.  Then  humanity's  native  curiosity  overcame 
the  terrors  of  narrow  theology.  Confidence  in  the  persistence 
of  the  order  of  the  universe  gained  ground  and  a  general 
interest  in  the  things  of  the  world  resulted.  The  spirit  of 
inquiry  soon  became  rife  and  with  it  came  a  healthy  scepticism. 
In  the  words  of  Machiavelli,  men  began  to  follow  the  real 
truth  of  things  rather  than  an  imaginary  view  of  them. 

The  mute  evidence  of  cathedral  churches  left  half  completed, 
or  with  one  spire,  or  none,  after  the  year  1400  or  1500,  testifies 
to  the  flow  of  human  enthusiasm  and  energy  toward  other 
channels.  That  so  many  mighty  cathedrals  could  be  con- 
structed in  Europe  from  900  to  1400  A.D.,  without  advance 
in  the  science  of  mechanics,  seems  remarkable.  But  their 
excellence  is  in  the  field  of  art  and  not  in  that  of  engineering. 

Close  acquaintance  with  them  reveals  to  the  engineer,  poor 
foundations,  cracked  arches,^  crooked  walls  and  leaning  towers^ 
and  settled  piers,'  quite  in  accord  with  the  annals*  of  failure 
and  collapse  which  is  the  history  of  their  construction.  In 
what  constituted  the  spirit  of  their  time,  in  imagination,  in 
fancy,  in  inversion  of  idea,  in  naivete  of  conception,  they  are 

»In  1284  the  central  tower  and  the  apse  vaulting  of  Beauvais  Cathedral 
collapsed  utterly.  The  dome  of  St.  Peter's  at  Rome  would  have  fallen 
long  since  but  for  the  iron  bandage  of  chains  placed  about  the  dome  in 
1742  by  Vanvitelli  under  the  direction  of  Poleni. 

^The  Campanile  at  Bologna  is  a  well-known  example. 

sAnnales  de  Sevilla,  1677,  "On  Dec.  28,  151 1.  a  split  pillar  (of  Seville 
Cathedral)  brought  down  all  the  central  tower  and  three  great  arches  with 
a  noise  that  stunned  the  city.  .  .  .  By  a  miracle  of  Our  Lady  of  the  Sea  it 
did  not  fall  at  once.  .  .  .  The  Archbishop  granted  indulgences  to  all  who 
would  assist  in  clearing  away  the  debris."  In  1890  it  collapsed  again. — The 
utter  and  complete  collapse  of  the  Campanile  of  San  Marco  at  Venice  in 
1902  is  recent  history. 

*See  Hamlin,  p.  197,  and  Feree,  Chronology  of  Cathedral  Churches  in 
France. — 'The  unscientific  Romanesque  vaulting,  etc.,  resulted  in  the  entire 
reconstruction  of  the  cathedrals  of  Bayeux  Bayonne,  Cambray,  Evreux, 
Laon,  Lisieux,  Le  Mans,  Noyon,  Poitiers,  Senlis,  Soissons  and  Troyes  about 
1200,"  etc. 


44  THE   SCIENCE   OF   MECHANICS. 

wonderful,  but  to  the  trained  eye  of  the  engineer,  the  method 
of  trial  and  blunder  through  which  they  were  achieved  is 
apparent. 

They  are  works  of  art  par  excellence,  there  is  little  science 
here,  except  the  experienced  skill  of  the  master  masons, 
whose  closely  guarded  guild  secrets  seem  to  have  been  trade 

/  tricks  rather  than  a  science  of  statics.     No  evidence  has  been 
discovered  tending  to  prove  that  the  cathedral  builders  had 

\  any  clear  conception  of  the  law  of  action  and  reaction,  or  of 
I  the  general  principle  of  moments,  but  they  may  have  used  a 
crude  method  of  determining  the  ratio  of  stresses  by  the  funi- 
cular method  of  using  weighted  strings  passing  over  pulleys. 
The  first  formal  exposition  of  this  method  seems  to  be  in  the 
works  of  the  Flemish  engineer  Stevinus  who  was  not  born  till 

v_  1548- 

The  opening  of  the  Renaissance  found  the  science  of  me- 
chanics not  very  much  further  advanced  than  where  Archi- 
medes had  left  it.  Now  men  began  to  study  and  speculate 
on  the  subject.  Most  eminent  and  successful  among  those 
who  so  occupied  themselves  are  the  following : 

1.  Copernicus  (1473-1543),  of  Thorn  in  Prussia,  who  set 
forth  the  system  of  astronomy  since  identified  with  his 
name  in  "De  Revolutionibus  Orbium  Coelestium."  He  main- 
tained that  the  sun  is  at  rest  and  that  the  planets  revolve 
about  it,  and  hinted  that  theology  and  mechanics  are  two 
distinct  branches  of  knowledge.  This  quotation  dimly  presag- 
ing the  law  of  gravitation  is  interesting:  "I  am  of  the  opinion 
that  gravity  is  nothing  more  than  a  natural  tendency  im- 
planted in  particles  by  the  Divine  Master  by  virtue  of  which, 
they  collecting  together  in  the  shape  of  a  sphere  do  form  their 
own  proper  unity  and  integrity.  And  it  is  also  to  be  assumed 
that  this  propensity  is  inherent  in  the  sun,  the  moon  and  the 
other  planets." 

2.  Leonardo  da  Vinci  (1452-1519),  the  Italian  painter, 
whose  manuscripts  give  a  crude  idea  of  the  statical  moment. 

3.  Peter  Ramus  (1515-1572),  who  contended  in  his  thesis 
for  the  Master's  degree  at  the  College  de  Navarre  that  all 
that  Aristotle  taught  was  false.     In  his  "Animadversiones  in 


4-  The  Contribution  of  Simon  Stevinus 
( 1 548-1 620). 

Simon  Stevinus  of  Bruges,  a  military  engineer  of  Prince 
Maurice  of  Orange  seems  to  have  been  a  man  of  genius  in 
experimental  research  as  well  as  in  practical  engineering.  His 
earliest  extant  work  is  the  "Beghinseln  der  Weegkonst"  pub- 
lished in  Dutch  at  Leyden  in  1586.  The  full  account  of  his 
researches  is  given  in  "Hypomnemata  Mathematica"  (Mathe- 
matical Memoranda)  a  large  volume  in  Latin  published  at 
Leyden  1608. 

This  volume  covers  in  six  books,  the  topics,  arithmetic, 
geometry,  cosmography,  practical  geometry,  statics,  optics  and 
fortifications.     The  division  on  Statics  treats  of, 

1.  The  elements  of  statics. 

2.  The  theory  of  center  of  gravity. 

3.  Practical  statics. 

4.  First  principles  of  hydrostatics. 

5.  Practical  hydrostatics.  , 

6.  Miscellaneous  topics. 

This  curious  medley  of  theory  and  practical  hints  was  no  doubt 
the  encyclopedia  of  mathematics  and  mechanics  of  the  period. 
A  revised  edition  in  French  was  published  by  Albert  Gerard  in 
1634.  Both  editions  are  very  fully  illustrated  with  wood  cuts. 
We  do  not  find  in  it  any  mention  of  dynamics.  Statics  is 
defined  as  the  interpretation  of  the  computations,  proportions 
and  conditions  of  equilibrium  (pondus)  and  of  weight  (gravi- 


THE   MEDIEVAL    PERIOD.  45 

Dialecticam  Aristotelis,"   1543,  he  strenuously  opposed  the 
scholastic  dogmas.  ^ 

4.  Guido  Ubaldi,  an  Italian,  who  published  in  his  "Mechani-     / 
corum  Liber"  (1577),  an  imperfect  idea  of  the  statical  moment.    I 

Of  this  period  is  the  work  of  two  of  the  great  contributors 
to  the  science  of  mechanics,  one  in  the  field  of  statics,  and  the 
other  in  the  field  of  dynamics,  of  which  he  was  the  founder — 
Simon  Stevinus,  an  engineer  of  Bruges  (1548-1620),  and 
Galileo,  a  professor  of  Florence  (1564-1642). 


46  THE   SCIENCE  OF  MECHANICS. 

tas).  The  weight  of  a  body  is  defined  as  its  (potentia  descen- 
sus in  dato  loco)  force  of  descent  in  a  given  place.  Center  of 
gravity  is  clearly  conceived  and  defined. 

Whereas  Archimedes  considered  only  the  action  of  parallel 
forces  at  right  angles  to  the  lever,  Stevinus  considers  the 
action  of  forces  in  any  direction  and  at  any  angle.  He  was 
the  first  to  give  a  solution  of  the  problem  of  stability  or  in- 
stability on  an  inclined  plane.  His  presentation  of  the  simple 
machines  differs  from  that  of  Archimedes  in  that  he  uses  the 
graphic  method  of  the  triangle  of  forces  in  the  solution  of  them. 

His  principal  contribution  to  the  science  is  this  idea  of  the 
parallelogram  or  triangle  of  forces  which  he  gives  by  many 
graphical  examples  without  definitely  proving  it  as  a  general 
principle  at  the  beginning.  It  was  not  completely  stated  and 
generally  admitted  as  a  principle  until  about  ninety  years 
later  when  Varignon  proved  it  geometrically  and  set  it  forth 
in  a  paper  before  the  Paris  Academy  (1687).  In  the  same  year 
Newton  and  Lami  also  published  a  proof. 

It  is  worthy  of  note  that  the  first  practical  exposition  of 
the  solution  of  engineering  problems  by  graphical  representa- 
tion of  forces  or  funicular  polygons,  now  so  commonly  in  use 
to-day  under  the  name  of  "graphic  statics"  was  published  by 
engineer  Stevinus,  about  three  hundred  years  ago. 

He  arrived  at  the  conception  of  the  triangle  of  forces  and 
the  conditions  of  stability  on  an  inclined  plane  by  his  famous 
"chain  of  balls  on  prism"  experiment.  This  is  given  in  the 
Hypomnemata  Mathematica  as  follows: 

/^I  Theorem.  Proposition  19.  If  a  plane  triangle  is  placed 
vertically  with  the  base  parallel  to  the  horizon,  and  upon  the 
other  two  sides  are  placed  single  globes  in  equilibrium  then 
according  as  the  right  side  of  the  triangle  is  to  the  left  so  is  the 
balancing  effect  of  the  left  globe  to  the  counterbalancing  effect 
of  the  right  globe.  -;,5 

Given:  Let  ABC  be  the  vertical  triangle  (Fig.^t)  with  base 
parallel  to  the  horizon  with  side  AB  double  BC,  ar^d  let  the 
globe  Aon  AB  be  of  equal  size  and  weight  to  that  £  on" 5 C. 

Question:  Demonstrate  to  us  that  as  AB  (2)  is  to  BC  (i) 
so  the  balancing  effect  of  globe  E  is  to  the  counterbalancing 
globe  D. 


THE  MEDI/EVAL   PERIOD.  47 

Construction.  Let  us  arrange  a  crown  of  fourteen  balls  of 
equal  size  and  weight  strung  together  at  equal  intervals,  and 
let  there  be  three  fixed  points  STV  which  are  touched  by  the 
string  so  to  admit  of  motion  of  ascent  or  descent  of  the  string 
of  balls. 

Demonstration:  If  the  balancing  effect  of  the  globes  DRQP 
is  not  equal  to  that  of  EF  they  must  be  heavier.  Suppose 
they  are,  then  ONML  being  equal  to  GHIK,  the  eight  globes 


Fig.  5. 
(From  Liber  Statica,  Vol.  IV,  p.  34.) 

D,  R,  Q,  P,  0,  N,  M  and  L  must  overbalance  the  six  E,  F,  G, 
H,  I  and  K  and  the  eight  will  go  down  and  the  six  will  rise  up. 
D  will  go  down  to  0  and  I  and  K  will  take  the  place  of  E  and 
F.  But,  if  this  is  so,  the  string  of  globes  will  now  be  situated 
as  before  and  by  the  same  cause  the  eight  globes  on  the  left 
will  go  down  and  the  six  on  the  right  will  go  up,  which  is 
saying  that  the  globes  of  themselves  produce  continual  and 
eternal  motion.  This  is  false.  Therefore  the  part  of  the 
string  DRQPNML  holds  the  part  EFGHIK  in  equilibrium. 
If  from  equal  things  equal  are  taken,  equals  remain,  therefore 
subtracting  ONML  and  GHIK,  DRPQ  balances  EF.  But 
four  being  held  in  equilibrium  by  two,  E  must  be  doubly  as 
effective  as  D.  Therefore  as  the  side  BA  (2)  is  to  the  side 
BC  (i)  so  the  balancing  effect  of  globe  E  is  the  counter- 
balancing effect  of  globe  D. 

As  a  corollary  it  follows  that  the  four  balls  and  the  two  balls 


48 


THE   SCIENCE   OF  MECHANICS. 


may  be  concentrated  in  globes  of  corresponding  magnitudes 
as  indicated  in  Fig.  ^.     Or  a  device  like  Fig.  ^  may  be  em- 
ployed. 3p']  3;o 
(..^o--     'From  this  Stevinus  comes  by  corollary  to  the  study  of  the 
condition  of  equilibrium  on  an  inclined  plane,  which  he  proves 


Fig.  6. 

(From  Liber  Statica,  Vol.  IV,  p.  36.) 


by  the  diagram,  Fig.  4.     This  diagram  is  the  earliest  exposition 
of  the  triangle  of  forces. 

He  then  generalizes  the  principle  for  practical  use,  in  Figs;, 
5  and  6,  where  CE  is  to  EO  as  the  weight  of  the  body  is  to  the/ 
pull  P.  From  this  principle  the  theory  of  the  funicular  polygon 
is  then  developed  as  indicated  in  Figs.  1^  and  ^. 

jji-        31:5 


Fig.  8. 


Fig. 


From  the  funicular  polygon  he  advanced  to  the  considera- 
tion of  the  conditions  of  statical  equilibrium  in  each  of  the 
simple  machines,  referring  back  to  his  proof  of  the  inclined 
plane.  Nowhere  does  he  state  the  principle  of  the  parallelo- 
gram of  forces  explicitly  as  a  general  rule  from  which  all  cases 
of  equilibrium  in  machines  may  be  deduced.  In  the  chapter 
on  practical  statics  the  simple  machines  are  fully  expounded 


THE  MEDIEVAL   PERIOD. 


49 


and  their  applications  indicated  in  illustrations  showing  cask 
being  moved  into  warehouses,  etc. 

The  chapter  on  hydrostatics  is  also  very  practical.     The 
weight  of  a  cubic  foot  of  water  at  Leyden  is  noted  (62  pounds) , 


Fig.  10. 


Fig.  II. 


(From  Liber  Statica,  Vol.  IV,  p.  162.) 

and  suggestions  on  ship  design  are  given.  It  is  a  question  as 
to  how  much  of  Archimedes'  Hydrostatics  was  known  to 
Stevinus,  but  the  probability  is  strong  that  Stevinus  dis- 
covered, or  at  least  proved  the  principle  of  Archimedes  by  his 


^ 


WW 


B 

Fig.  12.  Fig.  13.  Fig.  14. 

(From  de  Hydrostatices  Elementis,  p.  119.) 

own  method.  He  clearly  set  forth  for  the  first  time  the  fact 
that  the  pressure  of  a  liquid  is  independent  of  the  shape  of  the 
containing  vessel  and  depends  upon  the  height  and  area  of  the 
base.  His  method  of  reasoning  is  simple  and  convincing  and 
worthy  of  quotation. 


50  THE   SCIENCE   OF  MECHANICS. 

Suppose  a  mass  of  water  A  in  a  jar  of  still  water.  This  cube 
is  in  equilibrium.  For,  if  not,  let  us  suppose  it  descends,  then 
the  water  which  comes  into  its  place  must  also  descend  when 
it  comes  into  that  same  place  and  under  similar  conditions; 
but,  this  leads  to  perpetual  motion  which  is  absurd  and  con- 
trary to  our  experiences.  Therefore  the  cube  A  does  not  move 
down  nor  up.  It  is  in  equilibrium.  If  now  we  suppose  the 
surface  of  the  cube  A  to  become  solidified,  this  surface  or 
"vas  superficiarium"  will  be  subjected  to  the  same  circum- 
stances of  pressure. 

When  it  is  empty  it  will  suffer  an  upward  pressure  equal  to 
the  weight  of  the  absent  water  which  balanced  the  upward 
pressure.  If  we  fill  it  with  any  other  substance  of  any  specific 
gravity  it  is  plain  that  the  loss  in  weight  of  that  substance  in 
water  is  equal  to  this  same  upward  pressure  which  is  equal 
to  the  weight  of  the  water  displaced.  Figs.  9  and  10  illustrate 
experimental  proofs  with  cubes  of  specific  gravities  1-5  and 
4  times  that  of  the  fluid. 

Granted  that  the  pressure  on  the  base  of  a  cube  or  vertical 
parallelopiped  of  liquid  is  equal  to  its  weight,  by  following 
a  similar  method  of  imagining  portions  of  the  liquid  to  become 
solidified  or  to  be  cut  out,  Stevinus  shows  that  the  pressure 
on  the  base  of  a  vessel  is  independent  of  its  form,  and  proves 
the  laws  of  pressure  of  communicating  vessels  and  tubes. 

Perusal  of  Stevinus'  notes  indicates  that  he  had  a  hazy 
idea  of  the  principle  of  virtual  displacements.  He  had  ob- 
served that  what  a  simple  machine  gains  in  force  it  loses  in 
distance.  In  his  discussion  of  pulleys  he  notes  that,  "Ut 
spatium  agentis  ad  spatium  patientis,  sic  potentia  patientis 
ad  potentiam  agentis"  (Vol.  IV,  L.  3)  as  the  space  passed 
over  by  the  force  is  to  the  space  passed  over  by  the  resistance 
so  is  the  resisting  force  to  applied  force.  Here  he  strikes  close 
to  the  principle  of  work,  namely  that  in  a  perfect  machine  the 
product  of  the  force  and  distance  traversed  is  equal  to  the 
resistance  times  the  distance  through  which  it  is  overcome. 

We  have  here  in  Stevinus'  book  the  germ  of  the  idea  of 
virtual  displacements.  That  is,  if  in  a  simple  machine  we 
consider  any  virtual  or  possible  displacement  of  the  agent, 


THE  MEDIEVAL   PERIOD.  5I 

the  resistance  moves  over  a  corresponding  displacement  so 
that  in  every  case  the  product  of  the  acting  force  and  its  dis- 
placement is  equal  to  the  resisting  force  times  its  displace- 
ment. 

Stevinus  utilized  this  idea  in  a  narrow  limited  way,  applying 
it  in  the  calculations  on  the  simple  machines  but  not  attaining 
to  the  idea  of  work,  as  the  measure  of  force  acting  through 
distance,  nor  to  the  idea  of  the  balance  of  positive  and  negative 
work  in  a  machine.  His  chief  contributions  are  the  statical 
principle  of  the  triangle  of  forces,  the  founding  of  Graphic 
Statics,  and  the  exposition  of  the  conditions  of  buoyancy  and 
liquid  pressure.  While  he  was  developing  Statics  in  these 
directions,  his  young  contemporary  Galileo  had  been  experi- 
menting with  moving  bodies  and  was  laying  the  foundations 
of  Dynamics. 

REFERENCES. 

Ennemoser.     The  History  of  Magic. 

Rydberg.     Magic  in  the  Middle  Ages. 

Dollinger.     Studies  in  European  History. 

Adams,  B.     The  Law  of  Civilization  and  Decay. 

Whewell.     History  of  the  Inductive  Sciences. 

Melanchton.     Initia  Doctrinae  Physicse. 

Bacon.     Novum  Organum. 

Duhring.     Geschichte  der  Mechanik. 

Heller.     Geschichte  der  Physik. 

Eichen.     Mittelalterische  Weltanschauung. 

Schneider.     Geschichte  der  Alchemic. 

Figuier.     L'alchimie  et  les  Alchimistes. 

Cuvier.     Histoire  de  Sciences  Naturelles. 

Maury.     L'Antiquite  et  au  Moyen  Age. 

Fahie,  J.  J.     Galileo,  his  life  and  work. 

Vivian.     Life  of  Galileo. 

Boundry,  F.     Galilee,  sa  vie. 

Lord.     Beacon  Lights  of  History. 

Mach.     The  Science  of  Mechanics. 

Ball,  J.  W.     The  History  of  Mathematics. 

Cajori,  F.     A  History  of  Physics. 

Alberi.     Opere — Geo  lettore  Galilei,  16  vols. 

Mahafy.     Des  Cartes. 

Stevinus.     Hyponemata  Mathematica. 

Nasmith.     Pascal. 

Gerland  &  Traumiiller.     Geschichte  der  Physical  Experimentier  Kunst. 

Brewster.     Martyrs  of  Science. 

Lodge,  Sir.  O.      Pioneers  of  Science. 


52  the  science  of  mechanics. 

5.  The  Contribution  of  Galileo  Galilei 
(f^o/  (1564-1642). 

Taking  up  now  the  work  of  Galileo  we  find  that  he  caused 
a  revolution  in  mental  attitude  toward  the  study  of  natural 
phenomena.  The  Aristotelian  Natural  Philosophy  had  for 
centuries  been  regarded  as  an  infallible  authority  in  the  schools. 
In  1543,  Petrus  Ramus  (1515-1572),  a  scholar  of  the  University 
of  Paris,  was  forbidden  by  an  edict  of  Francis  I,  under  pain 
of  punishment,  to  teach  or  write  against  it.  To  Galileo,  a 
young  medical  student  of  noble  Florentine  family,  who  had 
come  to  disbelieve  in  the  dogmas  of  the  old  philosophy  belongs 
in  part  the  glory  of  emancipating  men's  mind  from  this  author- 
ity of  antiquity. 

Galileo  appealed  from  apriori  axioms,  presuppositions  and 
syllogistic  deductions  to  an  investigation  of  the  actual  facts. 
The  teachings  of  Aristotle  had  been  received,  "ipse  dixit," 
up  to  this  time,  in  spite  of  the  fact  that  some  of  them  were 
contradicted  by  daily  experience,  and  in  spite  of  the  fact  that 
easy,  simple  experiments  proved  them  wrong. 

To  quote  but  a  few  of  these  Aristotelian  notions  which 
were  blindly  accepted  and  believed — 

1.  Substances  were  divided  into  "corruptible"  and  "in- 
corruptible," chief  among  the  latter  were  the  heavenly  bodies. 

2.  Bodies  were  classified  as  absolute  heavy  bodies  and  ab- 
solute light  bodies  and  "sought  their  places";  the  light  bodies 
belonging  up  and  the  heavy  bodies  down. 

3.  Motions  were  classified  as  "natural  motions"  and  "violent 
motions." 

4.  Large  bodies  were  believed  to  fall  quicker  than  small 
ones,  or  the  velocity  of  falling  bodies  was  believed  to  be  in 
proportion  to  their  weight. 

Galileo  vigorously  attacked  this  Aristotelian  philosophy; 
he  appealed  from  authority  to  experiment,  to  nature.  He 
boldly  contradicted  the  teachings  of  Aristotle,  which  had  been 
accepted  and  believed  for  over  a  thousand  years.  By  direct 
experiment,  as  for  example,  by  dropping  weights  from  the 
leaning  tower  of  Pisa  he  proved  that  Aristotle  was  wrong. 


THE   MEDIEVAL   PERIOD.  53 

The  schoolmen  of  the  time  did  not  readily  relinquish  their 
errors  and  carried  on  long  and  bitter  controversies  with  him 
so  that  he  was  obliged  to  leave  Pisa  for  Padua. 

His  keenness  of  perception  is  well  illustrated  by  the  story 
of  the  discovery  of  the  isochronism  of  the  pendulum  through 
observations  on  the  gradually  decreasing  swing  of  a  hanging 
lamp  in  Pisa  Cathedral.  He  counted  his  pulse  as  the  lamp 
oscillated  over  a  smaller  and  smaller  arc  and  found  that  the 
number  remained  constant,  thus  verifying  his  suspicion  of 
isochronism.  He  is  also  credited  with  the  first  determination 
of  the  relation  between  the  time  and  length  of  a  pendulum 
and  the  application  of  it  in  a  metronome  for  the  use  of  physi- 
cians. A  scheme  for  a  pendulum  clock,  which  he  never  realized, 
is  found  among  his  manuscripts. 

Galileo  seems  to  have  been  the  first  to  set  forth  clearly : 

1 .  The  idea  of  force  as  a  mechanical  agent. 

2.  The  conception  of  mechanical  invariability  of  cause  and 
effect. 

3.  The  principle  of  the  independence  of  action  of  simul- 
taneous forces. 

The  rigorous  mechanical  explanation  of  motion  dates  from 
Galileo.  He  studied  carefully  the  motions  of  falling  bodies 
and  projectiles  and  found  their  laws  setting  forth: 

1.  All  bodies  fall  from  the  same  height  in  equal  times. 

2.  In  falling  the  final  velocities  are  proportional  to  the  times. 

3.  The  spaces  fallen  through  are  proportional  to  the  squares 
of  the  times. 

He  came  to  these  laws  experimentally  by  collecting  data 
on  the  time  of  descent,  the  final  velocities  and  the  distances 
traversed,  as  in  the  following  table,  g  being  a  constant. 

Time.  Velocity.  Space. 

1.  Ig  lXlg/2 

2.  2g  2X2g/2 

3.  32  3X3g/2 

4.  42  4X4g/2 
t                            tg  tX  tgl2 

The  experiment  on  grooved  planes  by  which  these  results 
were  obtained  are  now  well  known.  An  inspection  of  the 
table  shows  at  once  that  the  numbers  follow  the  simple  law, 


54  THE  SCIENCE  OF  MECHANICS. 

V  varies  as  gi, 

which  expresses  the  relation  between  the  first  and  second 
columns, 

5  varies  as  gt^/2, 

which  expresses  the  relation  between  column  one  and  column 
three,  while 

5  varies  as  v^/2g, 

is  the  relation  of  the  second  and  third  columns. 

The  first  two  of  these  expressions,  vccgt  and  socgt^/2  were 
used  by  Galileo  in  his  development  of  dynamics  to  the  neglect 
of  S(xv^/2g.  Later,  Huygens  took  up  the  expression  s<xv^l2g, 
and  made  important  advances  based  upon  it. 

It  was  observed  in  the  discussions  of  moving  bodies  after 
Galileo's  time  that  a  moving  body  had  a  certain  "efficacy" — 
that  there  was  inherent  in  a  moving  body,  something  that  cor- 
responds to  force.  Later  philosophers  debated  strenuously  as  to 
whether  this  efficacy  was  propottional  to  the  velocity  or  to  the 
velocity  squared.  But  it  will  be  perceived  from  an  inspection  of 
the  above  expressions  that  a  body  with  double  the  velocity  can 
overcome  a  given  force  through  double  the  time,  but  through 
four  times  the  distance.  With  respect  to  time,  therefore,  its 
efficacy  is  proportional  to  velocity;  but  with  respect  to  dis- 
tance, or  space  traversed,  its  efficacy  is  proportional  to  the 
velocity  squared. 

Before  the  time  of  Galileo,  force  was  treated  in  mechanics 
only  as  pressure;  after  his  time  the  ideals  of  force,  velocity 
and  acceleration  as  we  know  them  to-day  came  into  use.  That 
either  acceleration  of  motion  or  change  of  shape  is  the  imme- 
diate effect  of  force  is  the  fact  that  Galileo  perceived  and  set 
down  as  a  fundamental  and  invariable  rule  of  dynamics. 

He  determines  force  by  the  change  of  velocity,  or  the  ac- 
celeration it  produces,  and  he  may  be  said  to  have  discovered 
the  law  of  inertia  indirectly.  At  all  events,  his  conception  of 
dynamics  might  be  expressed  by  the  formula  F  =  m.a,  though 
he  did  not  so  express  it  because  his  conception  of  mass  was 
not  clear.  He  made  no  use  of  the  expression  .S  =  v^/2g,  which 
led  Huygens  to  his  conception  of  energy,  later  formulated  as 


THE   MEDIEVAL   PERIOD.  55 

F.S  =  mv^/2.  His  failure  to  do  so  appears  to  be  due  to  the 
same  cause — he  did  not  fully  grasp  the  modern  conception  of 
mass.  -  / 

In  Galileo's  "Delia  Scienza  Mecanica"  (1655),  Tom.  i,  p. 
265,  appears  the  first  clear  presentation  of  the  principle  of 
virtual  velocities.  This  idea  is  now  common  property  in  sev- 
eral forms,  one  of  which  is  the  familiar  dictum,  "what  is  gained 
in  speed  is  lost  in  power."  It  developed  slowly  into  the  law 
of  conservation. 

In  completer  form  it  is  now  stated  as  follows:  If  a  material 
system,  acted  on  by  any  forces  whatever,  be  in  equilibrium; 
and  we  conceive  the  system  to  experience,  consistently  with 
its  geometrical  relations,  any  indefinitely  small  arbitrary  dis- 
placement; the  sum  of  the  forces  multiplied  each  of  them  by 
the  resolved  part  parallel  to  its  direction,  of  the  space  described 
by  its  point  of  application,  will  be  equal  to  zero;  this  resolved 
part  being  considered  positive  when  it  lies  in  the  direction  of 
its  corresponding  force,  and  negative  when  in  an  opposite 
direction. 

Though  Guido  Ubaldi  in  his  "Mechanicorum  Liber"  called 
attention  to  the  idea  of  virtual  displacement  and  moments  in 
connection  with  the  lever,  and  though  Stevinus  makes  mention 
of  it,  Galileo  appears  to  have  been  the  first  to  apply  these 
ideas  to  all  the  simple  machines.  The  term  "moment"  of  a 
force  seems  to  have  meant  to  Galileo,  the  effort  that  tended 
to  set  a  machine  in  motion.  Therefore,  in  order  that  a 
machine  should  remain  at  rest  or  in  equilibrium  under  the 
action  of  two  forces  it  is  necessary  that  their  moments  balance. 
He  showed  that  the  moments  of  a  force  are  always  proportional 
to  the  force  times  its  virtual  velocity. 

In  his  "Mechanica,  sive  de  Motu,"  Wallis  uses  the  term 
moment  in  this  same  sense  and  bases  his  statics  on  the  equality 
of  moments  as  a  fundamental  principle. 

This  idea  is  most  prolific  and  many  later  writers  used  it  in 
varied  form  as  the  basis  of  their  formal  presentation  of  me- 
chanics Descartes  for  example  in  Lettre  73,  Tom.  i,  "de 
Mechanica  Tractatus"  (1657),  bases  his  whole  treatment  of 
Statics  on  a  single  principle  which  is  essentially  Galileo's  idea 


56  THE  SCIENCE  OF  MECHANICS. 

of  virtual  velocities.  His  conception  of  it  is,  that  it  requires 
exactly  the  same  energy  to  raise  a  weight  P  through  an  altitude 
.4,  as  a  weight  Q  through  an  altitude  B,  provided  that  P  is  to 
^  as  5  is  to  ^.  It  follows  from  this  that  any  two  weights 
attached  to  a  machine  will  be  in  equilibrium  when  they  are 
disposed  in  such  a  way  that  the  small  paths  they  can  simul- 
taneously describe  are  reciprocally  as  their  weights. 

The  same  idea  was  presented  in  another  aspect  by  Torricelli 
in  "De  Motu  Gravium  Naturaliter  descendentium"  (1644). 
His  conception  was,  that  when  any  two  weights  rigidly  con- 
nected together,  are  so  placed  that  the  center  of  gravity  is  in 
the  lowest  position  which  it  can  assume  consistently  with  the 
geometrical  conditions,  they  will  be  in  equilibrium.  Torri- 
celli's  principle  was  finally  presented  in  the  form, — any  system 
of  heavy  bodies  will  be  in  equilibrium  when  their  center  of 
gravity  is  in  its  lowest  or  highest  position.  His  presentation 
was  based  on  Galileo's  conception  of  virtual  velocities. 

Finally  a  century  later,  it  was  stated  about  as  we  have  it 
to-day  in  general  terms  by  John  Bernoulli,  in  his  letter  to  Varig- 
non  dated.  Bale,  Jan.  26,  1717,  published  in  "Nouvelle  Mecan- 
ique,"  Tom.  H,  sect.  9. 

These  ideas  gave  a  new  trend  to  the  development  of  mechan- 
ics. In  1743,  D'Alembert  built  the  first  Treatise  on  Dynamics 
on  this  principle.  Had  this  idea  of  virtual  displacements  been 
clearly  perceived  and  appreciated  by  Archimedes,  mechanics  as 
a  science  would  have  developed  much  more  rapidly.  This 
principle  is  of  such  universal  application  that  a  separate  rule 
is  no  longer  necessary  for  each  of  the  simple  machines;  it 
suffices  for  them  all. 

To  Bernoulli  belongs  the  credit  of  showing  that  the  prin- 
ciple of  virtual  displacements  may  be  made  the  basis  of  a 
whole  theory  of  equilibrium,  but  the  idea  originated  with 
Galileo.  He  also  applied  the  principle  in  his  "Discourse  on 
Floating  Bodies"  demonstrating  by  it  the  theory  of  buoyancy. 
In  spite  of  these  expositions  some  of  Galileo's  opponents  still 
held  blindly  to  the  Aristotelian  theory  that  the  breadth  or 
form  of  a  body  was  the  factor  that  determined  whether  it 
sank  or  floated. 


THE   MEDIAEVAL    PERIOD.  57 

In  general,  Galileo  prepared  and  familiarized  men's  minds 
with  the  correct  notion  of  interdependence  of  force  and  motion 
thus  clearing  the  way  for  the  generalizations  of  Newton's  Laws 
of  Motion.  Nowhere  does  he  state  these  laws  explicitly,  but 
their  perception  is  involved  in  the  solution  of  some  of  the 
dynamical  problems  in  his  books.  He  even  gives  two  of  the 
laws  of  motion  in  an  incomplete  way.  The  first  law  of  Newton 
is  a  generalization  of  Galileo's  theory  of  uniform  motion.  The 
second  law,  that  change  of  motion  is  due  to  force  and  is  pro- 
portional to  the  force  that  makes  the  change,  and  takes  place 
in  the  direction  of  the  force,  is  a  generalization  of  Galileo's 
theory  of  projectile  motion. 

Before  this  time,  it  was  commonly  believed  that  a  body 
could  not  be  affected  by  more  than  one  force  at  a  time,  and 
it  was  even  held  that  a  ball  shot  horizontally,  moved  in  a 
straight  line  until  the  force  was  spent  and  then  fell  vertically 
to  the  earth.  Galileo  demonstrated  in  his  fourth  Dialogue 
that  the  path  of  a  projectile  must  be  a  parabola,  the  resultant 
of  a  uniform  transverse  motion  and  a  uniformly  accelerated 
vertical  motion. 

He,  however,  did  not  attain  to  a  clear  discrimination  between 
mass  and  weight,  and  he  failed  to  see  that  acceleration  might 
be  made  a  means  of  measuring  the  magnitude  of  the  force  of 
gravity.  There  is  no  statement  of  the  third  law  of  motion, 
in  reference  to  action  and  reaction,  anywhere  in  Galileo's  work, 
though  there  is  a  suggestion  of  the  idea  of  it  in  some  of  his 
statements  in  the  "Delia  Scienza  Mecanica." 

Galileo  not  only  founded  Dynamics  but  he  made  perfectly 
clear  the  fact  that  force  may  produce  two  effects  upon  bodies, 
change  their  motion,  that  is  give  them  acceleration,  or  it  may 
change  their  form  or  shape,  that  is  deform  them.  In  his  study 
of  the  first  effect  he  developed  the  dynamical  laws  of  falling 
bodies,  of  projectiles  and  of  the  pendulum,  in  the  later  he 
founded  the  study  of  the  resistance  of  materials.  His  crude 
investigations  as  to  the  internal  structure  of  matter  and  his 
theory  of  its  deformation  and  resistance  in  the  form  of  col- 
umns, posts,  beams  and  cantilevers  is  set  forth  in  his  book 
"Discorsi  e  dimostrazioni   matematiche  intorno  a  due  nuove 


58  THE   SCIENCE  OF  MECHANICS. 

scienze,"  published  in  Leyden,  1638.  This  work  attracted 
little  or  no  attention  at  the  time  but  it  is  one  of  Galileo's  most 
substantial  contributions.  Although  he  wrote  some  sixteen 
volumes  in  all,  this  work  and  his  "Discorso  interno  alle  cose 
che  stanno  in  sur  I'acqua"  in  which  he  proves  the  static  law 
of  fluid  pressure,  contain  nearly  all  his  research  in  mechanics. 

After  Galileo's  time  we  no  longer  find  such  naive  and  obscure 
phraseology  as,  "motions  are  of  two  orders,  natural  and  vio- 
lent." Henceforth  the  notions  of  the  Aristotelians  became 
untenable.  Men  soon  came  to  recognize  that  all  bodies,  even 
the  heavenly  bodies,  were  probably  of  one  kind.  Force  came 
to  be  understood  as  that  which  causes  acceleration  in  a  body, 
or  deformation  in  a  body.  It  became  apparent  that  in- 
stantaneous and  continuous  forces  produce  unlike  effects,  and 
that  weight  is  a  continuous  force  drawing  bodies  toward  the 
earth.  A  little  later  the  great  Newton  explained  how  it  was 
that  all  bodies  fall  with  equal  velocities  from  the  same  height, 
barring  the  unequal  resistance  of  the  air. 

When  these  things  had  been  pointed  out  and  verified  by 
experiment,  the  foundati^on  of  the  study  of  moving  bodies  was 
laid  and  the  progress  ^f  Dynamics  was  sure  and  steady. 
Newton's  generalizations  followed  logically  upon  Galileo's  dis- 
cussions of  motion,  and  D'Alembert's  Treatise  on  Dynamics 
came  as  an  expansion  of  these  ideas.  It  is  no  exaggeration 
to  say  that  to  Galileo  we  owe  modern  mechanics.  Lagrange 
in  his  "Mecanique  Analytique"  testifies  to  Galileo's  greatness 
in  these  words: 

"Dynamics  is  the  science  of  forces  accelerating  or  retarding, 
and  of  the  various  movements  which  these  forces  can  produce. 
This  science  is  entirely  due  to  moderns,  and  Galileo  is  the  one 
who  laid  its  foundations.  Before  him  philosophers  considered 
the  forces  which  act  on  bodies  in  a  state  of  equilibrium  only; 
and  although  they  could  only  attribute  in  a  vague  way  the 
acceleration  of  heavy  bodies,  and  the  curvilinear  movements 
of  projectiles,  to  the  constant  action  of  gravity,  nobody  had 
yet  succeeded  in  determining  the  laws  of  these  daily  phe- 
nomena on  the  basis  of  a  cause  so  simple.  Galileo  made  the 
first  important  steps,  and  thereby  opened  a  way,  new  and 
immense,  to  the  advance  of  mechanics  as  a  Science. 


THE  MEDIEVAL  PERIOD.  59 

"These  discoveries  did  not  bring  to  him  while  living  as 
much  celebrity  as  those  which  he  had  made  in  the  heavens; 
but  to-day  his  work  in  mechanics  forms  the  most  soHd  and 
most  real  part  of  the  glory  of  this  great  man.  The  discovery 
of  Jupiter's  satellites,  of  the  phases  of  Venus,  of  the  Sun-spots, 
etc.,  required  only  a  telescope  and  assiduity;  but  it  required 
an  extraordinary  genius  to  unravel  the  laws  of  nature  in 
phenomena  which  one  has  always  under  the  eye,  but  the 
explanation  of  which,  nevertheless,  had  always  baffled  the 
researches  of  philosophers." 


PART  III. 

THE  MODERN  PERIOD,   1500  TO  1900. 

We  no  longer  believe  with  the  cave-men  that  thunder  is 
the  roar  of  an  angry  god,  nor  with  Luther  that  a  stone  thrown 
into  a  pond  will  cause  a  dreadful  storm  because  of  the  wrath 
of  devils  kept  in  prison  there;  but  we  still  believe  with  them 
that  wood  floats,  and  we  have  clear  ideas  of  the  conditions 
of  its  buoyancy.  The  characteristics  of  the  modern  period 
are  its  empiricism,  the  great  and  increasing  part  played  by 
natural  knowledge,  and  a  strong  conviction  of  the  importance 
of  sense  impressions  as  a  source  of  knowledge.  Added  to  this 
we  observe  an  enthusiasm  for  research  and  a  determination 
to  expose  error  regardless  of  controversy  or  consequences. 

Since  1700  the  whole  outlook  upon  the  universe  has  changed. 
Science  has  routed  the  old  theology,  and  altered  the  habits 
of  life  of  millions  by  its  influence  in  the  trades  and  industries. 
Though  there  is  still  nothing  more  mysterious  than  force,  the 
imps  of  that  weird  ante-world  of  Science  that  lurked  in  every 
zephyr  and  grinned  from  every  tree  and  dark  nook  are  now 
no  more.  Quite  apart  from  the  comforts  of  life  that  we  owe 
to  mechanics,  we  are  indebted  to  the  science  for  the  peace 
of  mind  which  a  rational  Natural  Philosophy  has  brought  us. 

Since  Galileo's  time  mechanics  has  been  characterized  by 
an  attitude  of  direct  experimental  inquiry  which  has  sought 
to  test  and  extend  the  conceptions  already  formed.  The 
science  has  grown  by  slow  expansion  and  accretion,  and  it  has 
often  been  some  time  before  new  conceptions  have  become 
susceptible  of  precise  statement.  It  seems  as  though  an  idea 
must  be  considered  and  turned  over  by  many  minds  before  it 
can  be  clearly  set  forth.  Even  the  great  masters  have  not 
always  presented  the  principles  which  they  have  contributed, 
in  the  form  in  which  we  now  state  them. 

Thus  it  is,  that  a  clear  statement  of  the  ideas  developed 
in  this  later  period  is  not  attained  to,  until  a  number  of 

60 


THE    MODERN    PERIOD.  6l 

workers  have  cultivated  the  field,  and  until  each  has  developed 
separately  facts,  which  when  correlated  and  worked  over,  by 
a  master  mind,  give  us  an  illuminating  view  of  the  whole 
subject. 

So,  in  passing  from  the  work  of  Galileo  to  that  of  the  next 
great  master  Christian  Huygens  (i  629-1 696),  we  pass  over 
a  number  of  men  whose  work  was  one  of  preparation,  ex- 
tension and  amplification,  rather  than  of  new  contribution. 
Such  were  these  workers  whose  activities  can  be  but  briefly 
referred  to  in  this  outline  of.  the  history  of  mechanics. 

Johann  Kepler,  1 571-1630,  of  Wiirtemburg,  Germany,  who 
in  "Astronomia  Nova,"  1609,  and  "Harmonice  Mundi,"  1619, 
set  forth  the  three  laws  of  planetary  motion,  viz: 

(i)  Each  planet  revolves  in  an  elliptic  orbit  having  the 
sun  as  its  focus;  (2)  the  straight  line  joining  the  sun  and 
planet  passes  over  equal  areas  in  equal  times;  (3)  the  square 
of  the  time  of  revolution  of  each  planet  is  proportional  to 
the  cube  of  its  mean  distance  from  the  sun.  Here  we  have 
the  statement  that  the  solar  system  is  disposed  according  to 
mathematical  and  mechanical  law. 

Francis  Bacon  (i 561-1626),  author  of  Novum  Organum 
(1620),  who  declared  for  science  on  the  ground  that  "knowledge 
is  power"  and  who  advocated  an  experimental  study  of  the 
world  with  a  view  to  improving  human  conditions. 

Marcus  Marci,  1595-1667,  published  at  Prague  in  1639 
"De  Proportione  Motus"  in  which  he  gives  correct  elementary 
notions  of  impact. 

Rene  Descartes,  1596-1650,  published  at  Amsterdam  in 
1644  his  "Principia  Philosophia;"  in  which  we  have  the  first 
notable  modern  endeavor  to  formulate  a  system  of  mechanics 
from  the  universal  point  of  view.  His  scheme  is  objectionable, 
but  he  called  attention  to  the  problem  of  a  universal  mechanical 
philosophy. 

Gilles  Personne  de  Roberval,  1 602-1 675,  published  in  the 
Memoirs  of  the  French  academy,  1668,  a  notable  paper  "Sur 
la  Composition  de  Movements." 


62  THE   SCIENCE  OF  MECHANICS. 

Otto  Von  Guericke,  1602-1686,  of  Magdeburg,  Germany,  an 
engineer  in  the  army  of  Gustavus  Adolphus  published  "De 
Vacuo  Spatio,"  1663,  and  "Experimenta  Nova,"  1672,  giving 
an  account  of  his  invention  of  the  air-pump,  1650,  and  various 
experiments  performed  with  it. 

Pierre  de  Fermat,  1601-1665,  of  Montauban,  France,  pub- 
lished between  1670  and  1680  a  series  of  monographs  on 
maxima  and  minima,  tangents,  curves,  centers  of  gravity 
and  copies  of  his  correspondence  with  Descartes,  Huygens 
and  Pascal  on  mechanical  problems,  under  the  title  "Opera 
Mathematica,"  which  cleared  the  way  for  later  advance. 

Evangelista  Torricelli,  1608-1647,  constructed  the  first 
mercurial  barometer  about  1643  and  applied  it  to  the  measure- 
ment of  variations  in  atmospheric  pressure. 

Edme  Mariotte,  c.  1620-1684,  who  in  his  "Trait^  du  Mouve- 
ment  des  Eaux,"  1686,  published  the  first  treatise  on  hydraulics 
and  advocated  and  developed  experimental  research  on  gravi- 
tation, hydraulics  and  pneumatics. 

Robert  Boyle,  1627-1691,  who  first  formulated  what  is  now 
known  as'^Buyles  Law,"  viz :  that  when  a  gas  is  at  a  constant 
temperature,  the  product  of  the  pressure  and  volume  remains 
constant,  howsoever  one  of  these  be  varied. 

Blaise  Pascal,  1 623-1 662,  who  published  his  studies  on  the 
question  of  fluid  pressure  in  "Recit  de  la  grande  experience  de 
I'equilibre  des  liquers"  (1648),  and  "Traite  de  I'equilibre  des 
liquers  et  de  la  presanteur  de  la  masse  de  I'air"  (1662).  One 
of  his  conclusions,  commonly  known  as  Pascal's  principle  is, 
that  "external  pressure  is  transmitted  by  fluids  in  all  directions 
without  change  in  the  intensity." 

All  of  these  investigators  either  set  forth  an  idea  of  some 
importance  or  simplified  and  extended  the  presentation  of 
accepted  ideas.  Later  investigators  were  familiar  with  their 
work  and  mounting  upon  it  attained  to  the  higher  conceptions 
of  the  science.  The  work  of  Kepler  in  astronomy  was  partic- 
ularly useful  to  Newton  in  attaining  to  his  grand  generalization 
of  the  law  of  universal  gravitation.  The  work  of  Descartes 
besides  presenting  a  highly  useful  method  of  combined  alge- 


THE  MODERN    PERIOD.  63 

braic  and  geometric  analysis,  suggested  the  idea  of  Conserva- 
tion, while  the  branches  of  hydrostatics  and  pneumatics 
could  not  have  progressed  far  without  the  researches  of 
Guericke,  Torricelli,  Mariotte,  Pascal  and  Boyle.  Their 
work  was  one  of  investigation,  experiment  and  study  in  rather 
narrow  fields.  It  was  a  work  of  preparation  upon  which  their 
successors  built  grandly. 


I.  The  Contribution  of  Christian  Huygens 
(1629- 1 696). 

Most  of  the  great  masters  of  mechanics  have  left  a  record 
of  practical  invention  as  well  as  of  theoretical  advance  in  the 
science.  Huygens  is  remembered  as  the  man  who  first  made 
a  good  clock.  Born  at  The  Hague,  and  educated  at  Leyden 
and  Breda,  where  he  studied  law  and  mathematics,  he  won 
fame  in  astronomy  as  well  as  in  mechanics.  In  1665  he  dis- 
covered the  rings  of  Saturn  with  a  telescope  which  he  had  con- 
structed. We  are  concerned  however  only  with  his  contri- 
bution to  mechanics.  He  carried  forward  Dynamics  by  de- 
veloping precise  statements  of  accelerated  motion  and  solving 
the  first  problems  in  the  dynamics  of  several  masses.  Galileo 
had  always  restricted  his  speculations  to  a  single  body. 

Huygens'  contributions  are  set  forth  in  his  publications; — 
"A  summary  account  of  the  laws  of  motion,"  Philosophical 
Transactions,  1669,  "Horologium  Oscillatorium,"  Paris,  1673, 
and  "Opuscula  Posthuma,"  Leyden,  1703.  The  complete 
mathematical  theory  of  the  pendulum  and  his  invention  of 
the  escapement  is  completely  set  forth  in  the  "Horologium" 
(1673),  which  is  a  work  worthy  to  rank  with  Newton's  Prin- 
cipia.  He  was  the  first  to  determine  the  acceleration  of  gravity 
by  the  pendulum,  and  also  the  first  to  enunciate  the  formula 
of  centrifugal  force,  F=mv^/r;  his  discovery  of  the  laws  of 
collision  of  elastic  bodies  was  announced  simultaneously 
with  that  of  Wallis  and  Wren. 

Huygens'  great  work  was  a  complete  exhaustive  theory  of 
the  pendulum  and  the  solution  of  the  problem  of  center  of 
oscillation.  He  considered  the  pendulum  to  be  made  of 
particles,  and  originated  the  idea  of  the  dynamics  of  more 


64  THE   SCIENCE  OF  MECHANICS. 

than  one  mass  or  particle.  He  employed  the  method  of  ag- 
gregating the  motions  of  particles,  but  though  he  used  symbols 
for  "moment  of  inertia"  and  "statical  moment,"  he  did  not 
define  and  assign  these  names.  Euler  appears  to  be  responsible 
for  the  term  moment  of  inertia  (Sw.r^).  This  method,  now 
so  general  in  its  application  in  the  mechanics  of  solids  and 
liquids  and  in  daily  use  by  engineers  and  physicists,  is  one  of 
the  great  inventions  of  Huygens.  It  gave  to  the  science  of 
mechanics  a  new  trend,  and  when  the  invention  of  the  calculus 
made  the  process  of  summation  easy,  this  method  brought  a 
wealth  of  progress  in  dynamics  and  hydrodynamics. 

In  the  dispute  as  to  whether  the  so-called  "efficacy"  of  a 
moving  body  is  proportional  to  the  first  or  second  power  of 
the  velocity,  Huygens,  who  had  originated  the  later  idea, 
maintained  it  strenuously.  The  dispute  was  really  one  of 
terms.  The  "efficacy"  of  a  moving  body  varies  as  its  velocity 
in  reference  to  the  time  and  as  the  square  of  the  velocity  in 
reference  to  the  space  passed  over.  Reference  to  the  time 
leads  to  what  Descartes  called  the  "quantity  of  motion" 
(momentum),  w.z?.  This  makes  the  notion  of  force  the  primary 
concept.  Reference  to  the  distance  passed  over  gives  the 
expression,  m.v^,  which  makes  work  or  energy  the  primary 
concept.  The  first  view  is  expressed  by  F.t  =  m.v  as  the  fun- 
damental equation  of  mechanics,  the  second  gives:  F.s  =  m.v^ 
as  the  fundamental  equation. 

In  1847,  Belanger  proposed  the  name  impulse  for  the  ex- 
pression F.t,  which  was  later  adopted  and  popularized  by 
Clerk  Maxwell  in  his  writing  on  matter  and  motion. 

Leibnitz  gave  the  name  "vis  viva"  to  the  expression  m-v"^ 
in  a  memoir  published  in  the  "Acta  Eruditorum,"  1695,  en- 
titled "Specimen  dynamicum  pro  admirandis  naturae  legibus 
circa  corporum  vires  et  mutuas  actiones  detengendis  et  ad 
suas  causas  revocandis,"  or  "A  dynamic  illustration  of  the 
astonishing  laws  of  the  power  of  bodies  in  their  reciprocal 
action  revealed  and  traced  back  to  their  causes."  He  in- 
tended, as  the  name  visa  viva  suggest,  to  indicate  a  measure 
of  the  force  of  a  body  in  actual  motion.  The  term  "vis 
motrix"  was  also  used  interchangeably  with  vis  viva  to  dis- 


THE   MODERN    PERIOD.  65 

tinguish  the  moving  force  from  statical  pressure  which  was 
called  "vis  mortua." 

The  difference  of  opinion  really  hinged  on  whether  force  or 
energy  is  to  be  considered  the  fundamental  notion.  Huygens 
and  his  party  maintained  the  latter  position,  while  Descartes 
and  later  Newton  accepted  force,  mass  and  momentum  as 
fundamental  notions.  This  dispute  went  on  for  fifty  years 
until  D'Alembert  in  1740  in  his  "Dynamique"  showed  it  to 
be  a  misunderstanding  as  to  terms,  not  facts. 

Later  Coriolis  (i  792-1 843)  introduced  the  more  common 
notation  of  ^w.t;^  for  vis  viva  or  kinetic  energy,  and  Poncelet 
adopted  the  same  plan,  but  the  conception  of  energy  we  owe 
then  to  Huygens. 

Another  of  his  important  achievements  was  the  solution 
of  the  problem  of  center  of  oscillation.  This  cannot  be  done 
without  recourse  to  the  new  method  which  he  used  so  suc- 
cessfully, namely  the  method  of  the  dynamics  of  particles. 
It  is  a  matter  of  every-day  observation  that  a  long  pendulum 
oscillates  more  slowly  than  a  shorter  one.  Therefore  if  we 
consider  the  component  particles  of  a  compound  pendulum 
as  so  many  simple  pendulums,  it  is  manifest  that  owing  to 
their  connections  they  all  vibrate  with  only  one  determinate 
period  of  oscillation.  There  must  exist  a  simple  pendulum 
that  has  the  same  time  of  oscillation  as  the  compound  pendulum. 
Its  length  measured  off  on  the  compound  pendulum  gives  us 
the  particle  or  point  that  preserves  the  same  period  of  oscil- 
lation as  if  it  were  detached  and  vibrating  as  a  simple  pendulum. 
This  point  is  the  center  of  oscillation. 

The  idea  which  Huygens  applied  in  the  solution  of  this  prob- 
lem is,  that  in  whatever  manner  the  particles  of  the  pendulum 
may  by  their  mutual  interaction  modify  each  other's  motions, 
in  every  case  the  velocities  acquired  in  the  descent  of  the  pen- 
dulum will  be  such  only  that  the  center  gravity  of  the  particles, 
whether  still  in  connection  or  with  their  connections  dissolved, 
is  able  to  rise  to  the  same  height  as  that  from  which  it  fell. 

Huygens'  proof  in  brief  is: — Let  OK  be  a  linear  pendulum 
made  up  of  a  large  number  of  masses  set  in  a  line  OK.  If  it 
be  set  free  it  will  swing  through  B  to  OK'  where  KX  =  XK'. 
s 


66 


THE   SCIENCE   OF  MECHANICS. 


The  center  of  gravity  will  ascend  just  as  high  on  the  second 
side  as  it  fell  on  the  first.  Now  if  at  OX  we  should  suddenly 
release  the  individual  masses  from  their  connections  the  masses 
could  by  virtue  of  the  velocities  impressed  upon  them  by  their 
connections,  only  attain  the  same  height  with  respect  to  center 


,-^*K' 


Fig.  15. 


Fig.  16. 


of  gravity.  If  the  free  outward-swinging  masses  be  arrested 
at  the  greatest  heights  they  attain,  the  shorter  pendulums 
will  be  found  below  the  line  0K\  the  longer  ones  will  have 
passed  beyond  it,  but  the  center  of  gravity  of  the  system  will 
be  found  on  OK'  in  its  former  position. 

The  enforced  velocities  are  proportional  to  the  distances 
from  the  axis;  therefore,  one  being  given  all  are  determined, 
and  the  height  of  ascent  of  the  center  of  gravity  may  be  found. 
Conversely  the  velocity  of  any  particle  is  determined  by  the 
known  height  of  the  center  of  gravity.  So  if  we  know  in  a 
pendulum  the  velocity  corresponding  to  a  given  distance  of 
descent  we  know  its  motion  is  defined.' 

If  now  on  a  compound  linear  pendulum  we  cut  off  the  portion 
L  equal  to  /,  and  if  the  pendulum  move  from  its  position  of 
greatest  displacement  to  the  position  of  equilibrium,  the  point 


THE  MODERN    PERIOD.  67 

at  the  distance  /  from  the  axis  falls  through  the  distance  K. 
The  masses  m,  m' ,  m"  at  the  distances  r,  r',  r",  will  fall  the 
distances  rk,  r'k,  r"k  .  .  .  and  the  distance  of  descent  of  the 
center  of  gravity  will  be: 

mrk-hin'r'k-\-ni"r"k-\- '  •  •  _     2wr 

If  now,  the  point  at  the  distance  /  acquires  on  passing  through 
the  point  of  Equilibrium  a  velocity  v,  the  height  of  ascent 
assuming  the  dissolution  of  connections  will  be  v^l2g  and  the 
heights  of  the  other  particles  will  be  (rvy/2g,  (r'vyi2g,  {r"vYl2g 
.  .  .  and  the  height  of  ascent  of  the  center  of  gravity  of  the 
liberated  masses  will  be: 

{rvY         Ar'vY  ,  m"(r"vy 
m +  m' 1 +  •  •  •        2^     2 

2g  2g  2g  _  v^Zmr^ 

m+m'+m"  2gLm' 

and 


2m        2gSw 


{a) 


But,  to  find  the  length  of  the  simple  pendulum  that  has  the 
same  period  of  oscillation  as  the  compound  pendulum,  it  is 
necessary  that  the  same  relation  must  exist  between  the  dis- 
tance of  its  descent  and  its  velocity  as  in  the  case  of  unimpeded 
fall.  If  y  is  the  length  of  this  pendulum,  ky  is  the  distance 
of  its  descent  and  vy  its  velocity.     Therefore, 

vy'^ 

—  =  ky 
2g 

or, 

y-=k.  (b) 

2g 

Multiplying  equation  (a)  by  equation  (&)  we  get 


y  = 


Xmr' 


Here  we  note  Huygens'  recognition  of  work  as  the  deter- 
minative of  velocity,  and  we  see  that  he  measures  it  in  terms 


68  THE   SCIENCE   OF   MECHANICS. 

of  the  second  power  of  the  velocity,  i.  e.,  V^.  This  work  he 
called  the  vis  viva  of  any  system  of  masses  such  as  m,  m' ,  m" , 
having  the  velocities  v,  v',  v"  as  expressed  in  the  formula. 

2  2  2 

Huygens  thus  clearly  indicated  that  the  center  of  gravity  is 
conserved  or  cannot  rise  higher  than  it  falls  and  in  establishing 
it  he  sets  forth  what  he  calls  the  principle  of  vis  viva,  or  the 
rule  that  the  work  or  energy  is  proportional  to  the  mass  times 
the  velocity  squared.  The  equation  for  work  and  energy 
F.s  =  m.v^  is  really  an  algebraic  statement  of  Newton's  second 
law  of  motion  and  is  fundamental  in  dynamics.  Lagrange 
based  his  work  on  it. 

The  writers  before  Huygens  did  not  have  a  clear  conception 
of  mass  as  distinguished  from  weight  nor  was  he  perfectly 
clear  on  this  point,  but  his  endeavors  to  explain  the  error  of 
the  pendulum  clock  of  the  Jean  Richer  expedition  to  Cayenne 
(167 1),  by  the  greater  centrifugal  force  at  the  equator,  show 
that  he  had  the  idea  of  mass  as  somehow  different  from 
weight. 

Huygens  may  then  be  said  to  have  contributed  the  mechan- 
ical principles  symbolized  by  those  type-expressions  and  their 
simple  derivatives, 
(i)  2mr2, 

(2)  F  =  mv'^/r, 

(3)  F.s'^m.v^. 

Thus  Huygens  originated  the  mathematical  method  by 
which  the  ideas  of  Galileo  were  applied  to  a  variety  of  problems. 
The  development  and  amplification  of  these  contributions 
by  their  successors  brought  a  wealth  of  progress.  Huygens' 
general  way  of  attacking  problems  of  masses  under  the  action 
of  forces  by  the  method  of  the  dynamics  of  a  particle  is  in 
almost  daily  use  by  the  physicist  and  the  engineer.  Like 
some  of  the  contributions  of  Archimedes,  the  contributions 
of  this  great  master  Huygens  have  an  eternal  value. 


THE  MODERN    PERIOD.  69 


REFERENCES— HuYGENS. 

(Euvres  completes  de  Christian  Huygens,  6  vols.  (1888  La  Haye). 

Christiani  Huygenii  de  circuli  magnitude  inventa. 

Christiani  Huygenii  Zuilichemii  Opera  Reliqua  (Amsterdam,  1728). 

Christiani  Huygenii  a  Zulichem  Opera  Varia,  1724. 

Christiani   Huygenii   Zulichemii  Opera   Mechanica;    Geometrica,   Astro- 

nomica  et  miscellanea,  1751. 
Christiani  Huygenii  Kosmotheros,   1698. 


2.  The  Contribution  of  Sir  Isaac  Newton 

(1642-1727). 

The  year  of  Galileo's  death  was  the  year  of  Newton's  birth. 
With  the  passing  of  the  great  Italian  scientist,  a  worthy 
successor,  the  greatest  of  all  experimental  philosophers,  was 
born  in  England.  He  established  order  in  the  domain  of 
Science  and  set  forth  the  great  laws  by  and  through  which 
mechanics  has  been  able  to  grow  and  prosper. 

Newton's  contribution  may  be  considered  under  two  head- 
ings: the  development  of  dynamics,  and  the  applications 
of  dynamics  to  the  great  problem  of  planetary  motions.  His 
work  is  a  logical  sequence  to  that  of  Stevinus,  Galileo,  Kepler 
and  Huygens.  The  "Philosophiae  Naturalis  Principia  Mathe- 
matica"  of  Newton  (1687),  commonly  called  the  Principia, 
is  one  of  the  most  extraordinary  products  of  human  genius, 
not  only  in  itself  but  in  the  revolution  which  it  effected  in 
theoretical  and  practical  mechanics.  In  it  we  find  much  more 
than  a  re-statement  of  the  general  principles  of  equilibrium, 
center  of  gravity  and  mechanical  powers  which  were  common 
property  at  this  time.  It  is  a  body  of  doctrine  based  upon  the 
contributions  of  all  preceding  inquirers  reduced  to  the  lowest 
terms. 

The  whole  body  of  doctrine  on  motion  of  projectiles  which 
had  been  developed  by  Galileo,  Huygens  and  others,  is  re- 
duced to  the  concise,  comprehensive  "axioms"  or  laws  of 
motion.  We  have  then  in  the  very  opening  pages  of  the 
Principia,  a  clarification,  precipitation  and  crystallization 
of  all  previous  contributions.  This  of  itself  was  a  great  gain, 
but  it  was  done  by  Newton  as  preliminary  to  further  advance, 
preliminary  to  dynamical  discussions  which  in  their  grand 


70  THE   SCIENCE  OF  MECHANICS. 

scope,  sweep  from  the  earth  to  the  planets  and  beyond  to 
the  utmost  limits  of  the  universe. 

The  "composition  of  forces"  he  indicates  as  a  corollary  to 

I  the  laws  of  motion.     This  was  a  new  idea.     In  Newton's 

point  of  view  if  a  body  in  space  is  acted  upon  by  an  impulsive 

force  it  moves  in  a  straight  line.     If  at  the  same  time,  another 

force  acts  upon  it  at  an  angle  inclined  to  the  first  force,  the 

body  takes  an  intermediate  course  called  a  "resultant"  path 

/  determined  by  drawing  the  diagonal  of  the  parallelogram,  the 

(_  two  sides  of  which  represent  the  magnitude  of  the  two  forces. 

The  truth  of  this  principle  is  made  to  depend  upon  the  laws 
of  motion.  From  being  a  mere  statement  of  experience,  as 
it  was  with  Galileo  and  Stevinus  the  parallelogram  of  forces 
is  correlated  to  the  fundamental  laws  of  motion  and  deduced 
as  following  at  once  from  them.  What  Galileo  and  Stevinus 
said  in  pages,  Newton  said  in  a  paragraph.  The  principle 
of  the  parallelogram  of  forces  is  not  merely  stated,  it  is  deduced 
from  three  fundamental  laws  of  motion. 

On  the  same  axioms  Newton  based  the  whole  theory  of 
central  forces.  He  supposes  a  body  to  be  acted  upon  by 
two  forces  as  above,  but  supposes  that  the  second  force  acts 
on  it  in  a  new  direction  in  succeeding  instants.  Then  the 
successive  diagonals  of  the  parallelograms  of  the  forces  will 
be  successive  sides  of  a  polygonal  figure  and  the  lines  of  the 
deflecting  forces  will  cut  one  another  within  the  figure.  If 
they  meet  in  a  point  forming  a  series  of  triangles  of  equal 
area  it  is  easy  to  see  that  the  path  of  the  body  is  the  same  as 
though  a  single  force  acted  upon  the  body  to  produce  motion 
forever  in  a  straight  line  and  a  second  force  acted  upon  it  to 
deflect  it  continually  to  a  point  within  the  polygon.  The  limit 
of  such  a  path,  as  the  polygonal  sides  become  smaller,  is  a 
curve. 

In  this  manner  centrifugal  forces  and  curvilinear  motion 
are  demonstrated  and  their  laws  set  forth.  Thus  a  whole 
system  of  dynamics  is  developed  from  the  geometrical  and 
mathematical  relations  of  diagrams  of  the  parallelogram  of 
forces.  Newton  founded  the  correct  theory  of  motion  about  a 
center  and  the  whole  system  of  dynamics  involved  with  it. 


THE   MODERN   PERIOD.  7 1 

He  set  forth  the  principle  of  equal  areas  described  in  equal 
times  as  the  test  of  a  central  force  and  gave  the  mathematical 
proof. 

A  body  exposed  to  the  action  of  a  central  force  and  given 
an  impulse  in  a  straight  line  will  neither  fall  toward  the  central 
force  nor  proceed  in  a  straight  line,  but  will  take  an  inter- 
mediate diagonal  or  curvilinear  path.  Newton  brought  to 
bear  on  this  problem  all  the  geometrical  and  mathematical 
knowledge  of  his  day,  as  well  as  his  own  fluxional  calculus, 
and  established  the  theorem  that  a  body  projected  in  a  straight 
line  and  subjected  to  the  action  of  a  central  force  will  revolve 
in  some  one  of  the  conic  sections  if  the  force  vary  inversely 
as  the  square  of  the  distance  from  the  focus, — which  of  the 
conic  sections,  he  shows,  depends  on  the  ratio  of  the  forces. 
This  dynamical  theorem  is  the  starting  point  of  Newton's 
system  of  celestial  mechanics. 

A  variety  of  consequences  follows  mathematically  from  this 
theorem.  The  dynamics  of  elliptic  orbits  is  established  at 
once  upon  a  sound  basis  and  the  interrelation  of  the  functions 
of  motion  follow  inevitably. 

He  also  took  up  the  abstract  theory  of  the  attractions  which 
portions  of  matter  may  be  conceived  to  exert  upon  each  other, 
showing  that  if  the  particles  be  attracted  according  to  the 
law  of  inverse  square  of  the  distance  and  if  they  be  aggregated 
into  spherical  masses  these  spheres  will  themselves  attract 
accordingly  to  the  same  law,  and  that  the  attraction  would  be 
directed  to  the  centers  of  the  spheres  and  be  proportional  to 
the  matter  contained  in  them,  divided  by  the  square  of  the 
distance  between  the  centers. 

From  this,  to  the  law  of  universal  gravitation  seems  but  a 
step.  But  though  Newton  is  said  to  have  entertained  this 
theory  as  early  as  1666,  it  was  not  till  1672,  when  the  data  of 
Picard  on  the  figure  of  the  earth  were  obtained,  that  Newton 
justified  it  and  became  convinced  of  its  truth.  He  first  gave 
it  out  in  his  lectures  in  1684.  It  was  published  latter  in  his 
treatise  "De  Motu"  and  in  the  "Principia."  In  Book  III, 
Proposition  4  of  the  latter,  he  calculates  the  acceleration  of 
the  moon  toward  the  earth  and  shows  that  starting  from 


72  THE   SCIENCE  OF  MECHANICS. 

rest  with  this  acceleration,  it  would  fall  towards  the  earth 
i6  feet  in  the  first  minute  and  that  at  the  earth's  surface  60  times 
nearer,  the  same  distance  would  here  be  fallen  through  in 
one  second  which  was  almost  exactly  the  value  obtained  by 
Huygens  in  his  experiments. 

The  action  of  force  without  a  medium  to  transmit  it,  appears 
to  have  troubled  Newton,  and  this  is  not  surprising  when  one 
considers  that  the  pull  of  the  sun  on  the  earth  is  equal  to  a 
force  sufficient  to  break  a  million  million  round  steel  rods 
each  twenty-five  feet  in  diameter. 

The  law  of  universal  gravitation  is  the  basic  principle  of 
Newton's  applications  of  his  dynamics  to  planetary  motions; 
when  he  had  achieved  it,  all  the  mechanism  of  the  universe 
lay  like  an  open  book  before  him.  It  was  now  possible  to 
apply  mathematical  analysis  with  absolute  precision  to  the 
problems  of  astronomy. 

At  once  many  new  conceptions  came  into  view.  Neither 
Galileo  nor  Huygens  had  clearly  distinguished  mass  from 
weight,  but  now  it  followed  at  once,  that  the  same  body  must 
have  a  different  weight  at  different  places  on  the  surface  of 
the  earth,  and  might  even  be  conceived  imder  certain  condi- 
tions to  have  no  weight.  We  arrive  now  for  the  first  time  at 
a  clear  idea  of  mass.  The  idea  of  force  ^s  first  propounded  by 
Galileo  was  now  seen  to  be  of  universal  application.  Finally 
the  law  of  action  and  reaction  was  clearly  stated  and  set  forth. 
These  are  most  illuminating  conceptions. 

It  began  to  be  evident  after  this,  that  gravity  was  a  force 
measurable  like  any  force  in  terms  of  mass  and  acceleration, 
though  it  was  some  time  before  the  principle  was  stated  in 
the  concise  algebraic  form, 

F=m.a, 
W=m.g, 

Newton's  concept  of  mass  is  in  fact  the  corner-stone  of  his 
dynamics. 

At  the  beginning  of  the  Principia  we  find  a  series  of  funda- 
mental conceptions  given  in  a  series  of  definitions.  The  final 
one  refers  to  mass,  as  follows: 


THE   MODERN    PERIOD.  73 

Definition  I.  (As  to  mass.)  "The  quantity  of  any  matter 
is  the  measure  of  it  by  its  density  and  volume  conjointly. 
This  quantity  is  what  I  shall  understand  by  mass  of  a  body  in 
the  discussion  below.  It  is  ascertainable  from  the  weight  of 
the  body  for  I  have  found  by  pendulum  experiments  of  high 
precision,  that  the  mass  of  a  body  is  proportional  to  its  weight, 
as  will  hereafter  be  shown. 

Definition  II.  "Quantity  of  Motion"  is  the  measure  of 
it  by  its  velocity  and  quantity  of  matter  conjointly. 

Definition  III.  The  resident  force  "vis  insita,"  i.  e., 
inertia  of  matter  is  a  power  of  resisting,  by  which  every  body, 
so  far  as  in  it  lies,  persists  in  its  state  of  rest  or  of  motion  in  a 
straight  line. 

Definition  IV.  "An  impressed  force  is  any  action  which 
changes  or  tends  to  change  the  state  of  rest  or  of  uniform 
motion  in  a  straight  line."  This  defines  force  as  the  cause 
of  acceleration  or  tendency  to  acceleration  of  a  body. 

The  laws  of  motion  as  Newton  enunciates  them  are: 

Law  I.  Every  body  persists  in  its  state  of  rest  or  of  uni- 
form motion  in  a  straight  line,  except  in  so  far  as  it  is  com- 
pelled to  change  that  state  by  Impressed  forces. 

Law  II.  Change  of  motion  {i.  e.,  of  momentum)  is  pro- 
portional to  the  moving  force  impressed,  and  takes  place  in 
the  direction  of  the  straight  line  in  which  such  is  impressed. 

Law  III.  "Reaction  is  always  equal  and  opposite  to 
action,  that  is  to  say  the  actions  of  two  bodies  upon  each 
other  are  always  equal  and  directly  opposite." 

To  these  are  added  a  number  of  corollaries.  The  first 
and  second  relate  to  the  principle  of  the  parallelogram  of 
forces,  the  others  are  logical  consequences  of  the  laws.  Then 
follow  the  propositions  in  two  books,  the  first  treating  of  the 
motion  of  bodies  in  non-resisting  media  and  the  second  in 
resisting  media. 

The  work  of  Newton  may  be  summed  up  in  his  definitions 
and  laws.  The  great  result  of  his  work  was  the  clear  concept 
of  mass  and  the  conception  that  bodies  mutually  cause  ac- 
celeration in  each  other  dependent  upon  space  and  material 
circumstances. 


74  THE   SCIENCE  OF  MECHANICS. 

Mach^  sums  the  matter  up  by  saying  that  in  reality  only 
one  great  fact  was  established,  viz:  that  "different  pairs  of 
bodies  determine  independently  of  each  other,  and  mutually, 
in  themselves,  pairs  of  accelerations  whose  terms  exhibit  a 
constant  ratio,  the  criterion  and  characteristic  of  each  pair." 

Newton  did  not  state  his  results  in  algebraic  terms.  He 
developed  the  deductions  in  the  "Principia"  geometrically 
though  there  seems  to  be  no  doubt  but  that  many  of  his  con- 
clusions were  arrived  at,  and  perhaps  first  proven  by  his 
method  of  fluxions  or  fluxional  calculus. 

Had  Newton  stated  his  principles  algebraically,  the  formula 
for  force,  F=m.a  would  probably  have  been  expressed  by 
him  as  F=in.v/t  or  F.t  =  m.v.  Acceleration  he  would  have 
expressed  as  the  time  rate  of  change  of  velocity  vjt,  and  forces 
he  would  have  measured  by  their  momentum,  mXv.  Up  to 
Newton's  time  such  formal  presentation  of  the  Science  of 
Mechanics  as  there  was,  had  been  made  on  the  geometrical 
method  and  Newton  following  in  the  steps  of  Archimedes, 
Stevinus  and  Galileo,  used  this  method  in  his  "Principia," 
representing  forces  by  lines  by  the  graphical  method. 

The  fact  that  acceleration  may  also  be  expressed  as  a  space 
rate  of  velocity  squared,  or  analytically  by  the  formula 
a  =  v^l2s  is  not  apparent  geometrically.  Substitution  in  the 
algebraic  formula  F=m.a  gives  F.s  —  m.v'^l2,  or  the  expression 
for  work  and  energy  which  is  not  considered  in  Newton's 
geometrical  analysis,  and  which  was  the  point  of  view  from 
which  Descartes,  Huygens  and  Leibnitz  approached  the 
subject. 

This  difference  in  point  of  view  gave  rise  to  the  long  con- 
troversy already  mentioned  between  the  English  disciples  of 
Newton  and  the  continental  school  or  the  adherents  of  Huygens 
and  Leibnitz.  The  so-called  Galileo-Newtonian  school  main- 
taining that  momentum  {F.t  =  m.v)  was  the  only  correct 
measure  of  force  and  the  Leibnitzian-Huygenian  school  main- 
tained with  equal  vigor  that  force  was  a  function  of  the  "vis- 

viva    or  energy  I  F-  5  =  — —  I  • 

iDr.  E.  Mach,  Science  of  Mechanics. 


THE   MODERN   PERIOD.  75 

For  fifty  years  mechanics  developed  along  these  two  sep- 
arate paths.  The  English  investigators  long  followed  the 
formal  geometrical  presentation  of  Newton,  and  the  French, 
Germans  and  Swiss  developed  their  mechanics  on  the  work  of 
Huygens,  using  the  calculus  of  Leibnitz.  It  was  not  till 
mathematical  analysis  came  to  be  applied  to  mechanics  and  an 
analytical  scheme  was  developed  by  D'Alembert  in  his  "Traite 
de  Dynamique"  (1743)  that  they  were  reconciled,  and  brought 
into  accord. 

It  was  then  seen  that  both  can  be  derived  from  Newton's 
fundamental  equation  F=m.a.  If  the  acceleration  is  meas- 
ured as  a  time  rate  of  velocity,  we  get  a  —  v/t  and  if  the  accel- 
eration is  measured  as  a  space  rate  of  the  velocity  squared 
we  get  a  =  v^/s,  which  are  of  the  same  dimensions. 

Newton  made  very  clear  the  conception  that  the  effect 
of  a  force  is  to  change  the  size  or  shape  of  a  body  or  to  change 
its  velocity,  that  is  to  give  it  acceleration.  For  studying  the 
flux  or  flowing  relations  of  quantities  he  devised  his  "Method 
of  Fluxions"  now  commonly  known  as  the  calculus.  The 
basic  idea  of  his  Fluxions  is  this.  He  considers  a  "fluent"  as 
a  quantity  that  gradually  and  indefinitely  increases  or  flows. 
The  velocities  at  which  such  fluents  move  he  defines  as  fluxions 
("Quas  Velocitates  appello  Fluxiones,  aut  simpliciter  Veloci- 
tates  vel  Celeritates").  With  the  development  of  this  method 
we  have  at  hand  an  instrument  for  tracing  changing  phenomena 
by  the  relation  or  ratio  of  elements.  This  is  an  invention 
of  inexpressible  value  to  Mechanics.^  The  method  appears 
to  have  been  first  used  by  Newton  as  early  as  1666  and  is 
found  in  his  MS.  "De  Analysi  per  Equationes  Numero 
Terminorum  Infinitas,"  which  was  given  to  his  students  in 
1669. 

Sir  Isaac  Newton  is  to  be  credited  then  with  a  general 
clarification  and  formulation  of  the  investigations  of  all  his 
predecessors,  and  these  specific  contributions: 

1.  The  concept  of  mass. 

2.  The  generalization  of  the  idea  of  force. 

'Prof.  John  Perry's  "Calculus  for  Engineers"  exemplifies  the  practical 
value  of  the  Calculus  of  Newton. 


76  THE   SCIENCE  OF  MECHANICS. 

3.  The  laws  of  motion. 

4.  The  theory  of  central  forces. 

5.  The  theory  of  attraction. 

6.  The   system   of  dynamics   based   on   the   conception 

F.t  =  M.v. 

7.  The  method  of  fluxions  or  the  fluxional  calculus. 

8.  The  law  of  universal  gravitation,  and  the  application 

of  his  abstract  dynamics  to  planetary  motions. 

Before  Newton  the  science  consisted  of  the  more  apparent 
rules  of  statics  as  developed  by  Archimedes  and  Stevinus 
and  the  uncorrelated  principles  of  dynamics  as  worked  out  by 
Galileo  and  Huygens.  Newton  reduced  these  cumbersome 
unconnected  rules  of  statics  and  dynamics  to  three  formal  laws 
of  motion  and  founded  a  system  of  dynamics  of  universal 
application  which  has  been  found  all-sufficient  to  co-ordinate 
the  mechanical  phenomena  of  the  universe. 

In  cqnclusion  we  may  say  that  the  principles  formulated  by 
Newton  cover  all  statical  and  dynamical  problems.  Much  of 
the  work  of  later  masters  has  been  a  verification  and  an  exten- 
sion of  the  work  begun  by  him.  The  scienc  of  mechanics,  as 
now  generally  taught,  is  founded  upon  them. 

Playfair  says^  in  his  dissertation  on  Newton,  "No  one  ever 
left  knowledge  in  a  state  so  different  from  that  in  which  he 
found  it.  Men  were  instructed  not  only  in  new  truths,  but  in 
new  methods  of  discovering  truth ;  they  were  made  acquainted 
with  the  great  principle  which  connects  together  the  most 
distant  regions  of  space  as  well  as  the  most  remote  periods  of 
duration  and  which  was  to  lead  to  future  discoveries,  far  beyond 
what  the  wisest  or  most  sanguine  could  anticipate." 

REFERENCES— Newton. 

Brewster,  D.     Memoirs  of  Sir  Isaac  Newton.     (2  vols.) 

Ball,  W.  R.     An  essay  on  Newton's  Principia. 

Newton.     Principia. 

Mach,  E.     The  Science  of  Mechanics. 

Ball,  W.  R.     A  Short  History  of  Mathematics. 

Cittenden.     Life  of  Newton. 

iPage  133,  Dissertation  11,  Playf air's  "Progress  of  Math,  and  Phys. 
Sciences,"  1820. 


THE   MODERN   PERIOD.  77 

Hosley,  Opera  Omnia,  1779. 

Maxwell.     Matter  and  Motion. 

Thomson  &  Trait.     Natural  Philosophy. 

Cajori,  F.     A  History  of  Physics. 

Hersley's  Newton.     London,   1785. 

Pearson,  C.     The  Grammar  of  Science. 

Pemberton.     View  of  Newton's  Philosophy. 

Playfair.     Progress  of  Mathematical  and  Physical  Sciences,  1820. 

Perry,  J.     Calculus  for  Engineers. 

3.  The  Contributions  of  Varignon,  Leibnitz,  the 
Bernoullis,  Euler  and  D'Alembert. 

Pierre  Varignon  (1654-1722).  S 

In  the  same  year  that  Newton  pubHshed  his  Principia  (1687), 
there  was  presented  before  the  Paris  Academy  a  work  on 
Statics  by  Pierre  Varignon  based  on  the  principle  of  moments 
which  he  developed  geometrically  from  the  parallelogram  of 
forces. 

The  book  was  published  after  his  death  under  the  title, 
"Project  d'une  Nouvelle  Mecanique"  with  the  dedication, 
"Illustrissimo  clarrissimoque  viro  D.  D.  Isaaco  Newton."  It 
begins,  "La  Mecanique  en  general  est  la  Science  du  Mouve- 
ment,  de  la  cause  de  ses  effects;  en  un  mot  de  tout  ce  qui  y  a 
rapport.  Par  consequent  elle  est  aussi  la  science  de  proprietez 
et  des  usuages  de  Machines  ou  Instruments  propres  a  faciliter 
le  mouvement;"  i.  e.,  "Mechanics  is  in  general  the  science  of 
motion,  of  its  cause  and  of  its  effects;  in  a  word  of  all  that 
pertains  to  motion.  Consequently  it  is  also  the  science  of 
machines."  We  meet  here  in  Varignon's  book  a  system  of 
mechanics  which  is  essentially  dynamical,  including  statics 
as  the  special  case  where  forces  counterbalance. 

After  defining  mechanics  thus,  as  the  science  of  motion  and 
the  theory  of  machines,  he  says  this  treatise  will  be  divided 
into  ten    sections: 

(i)  Axioms,  postulates  and  propositions; 

(2)  Weights  suspended  or  supported  by  strings; 

(3)  Pulleys; 

(4)  Wheel  and  axle; 

(5)  The  lever; 


78  THE   SCIENCE   OF  MECHANICS. 

(6)  The  inclined  plane; 

(7)  The  screw; 

(8)  The  wedge; 

(9)  The  general  principle  of  the  simple  machines; 

(10)  The  equilibrium  of  fluids. 

The  first  section  treats  of  the  definitions,  axioms  and  hy- 
potheses upon  which  the  work  is  based.  The  idea  that  forces 
may  act  each  upon  other  and  maintain  a  body  at  rest  is 
emphasized.  Then  the  suppositions  are  made  that  in  the 
geometrical  treatment  of  machines  the  parts  are  to  be  con- 
sidered as  without  weight  and  friction,  perfectly  mobile  upon 
their  axes,  cords  are  to  be  considered  as  perfectly  flexible 
without  weight,  without  elasticity  and  without  stretch  or 
elongation. 

The  principle  of  the  parallelogram  of  velocities  is  now  stated 
geometrically  as, 

Lemma  I. 

In  order  to  help  the  mind  to  conceive  compounded  motions 
let  us  conceive  the  point  A  without  weight  to  move  toward  B 
along  the  line  AB,  and  at  the  same  time  suppose  that  the 
line  itself  moves  uniformly  towards  CD  along  AC  remaining 
always  parallel  to  itself,  that  is  to  say  maintaining  always 
the  same  angle  with  AC.  Of  these  two  movements  com- 
mencing at  the  same  time  let  the  velocity  of  the  first  and  of 
the  second  be  as  the  sides  AB  and  ^C  of  the  parallelogram 
A  BCD.  Then  in  the  parallelogram,  I  say  that  by  the  action 
of  the  two  forces  upon  A,  this  point  will  travel  along  the  di- 
agonal AD  oi  the  parallelogram  during  the  time  that  AB  and 
AC  are  being  traversed. 

Lemma  II. 

If  the  point  A  without  weight  is  pushed  in  the  same  time 
and  uniformly  by  two  forces  E  and  F  acting  upon  it,  along 
the  lines  AC  and  AB  acting  at  the  angle  CAB.  The  united 
action  of  these  two  forces  will  move  A  along  the  diagonal 
of  the  parallelogram  AD  in  the  same  time  that  A  would  move 
to  C  or  to  B  and  as  though  a  force  in  the  proportion  ol  AD  to 
CA  or  AB  had  acted  upon  it. 


THE   MODERN    PERIOD. 


79 


The  parallelogram  of  velocities  and  of  forces  is  here  con- 
ceived as  axiomatic.  On  these  conceptions  Varignon  builds 
up  a  logical  geometrical  development  of  mechanics.  He  dem- 
onstrates the  principle  of  statical  moments  by  a  geometrical 
theorem  in  which  he  shows  that  the  product  of  a  force  F 
(represented  graphically  by  a  line),  times  its  lever  arm,  a 


X  /\ 


Fig.  17.  Fig.  18. 

(Diagram  from  "Nouvelle  Mecanique.") 

(another  line),  the  product  of  which  is  a  certain  area,  is  equal 
to  the  complementary  moment  also  represented  by  an  equal 
area  in  the  diagram.  For  example,  in  the  diagram.  Fig.  12, 
from  the  Nouvelle  Mecanique  we  have  then  F  X  a  -{-  F'  X  c 
=  R  X  b  or  the  moment  of  the  diagonal  of  a  parallelogram  of 
forces  is  equal  to  the  sum  of  the  moments  of  the  other  two 
sides.  The  point  O  may  be  chosen  either  without  the  paral- 
lelogram within  it  or  on  one  of  the  sides.  Varignon  demon- 
strates all  three  cases  by  proving  that  the  areas  are  equal  by 
geometry. 

If  the  point  0  be  taken  within  the  parallelogram  and  the 
perpendiculars  be  then  drawn  we  have  FXa—  F'Xc  = 
R  X  b.  Finally  if  0  be  taken  on  the  diagonal  the  moment  of 
the  diagonal  is  zero  and  we  have  Fa  =  F'c.     In  every  case  the 


8o 


THE   SCIENCE   OF  MECHANICS. 


proof  consists  in  a  geometrical  proof  of  equality  of  areas,  the 
area  being  the  representation  of  the  product  of  a  line,  the  lever 
arm,  by  another  line  representing  the  force.  The  principle 
thus  proven  is  often  called  Varignon's  principle.  It  is  hardly 
credible  that  this  principle  of  moments  was  not  established 
until  1687,  but  such  appears  to  be  the  fact.     The  proof  of 


Fig.  19. 


Fig.  20. 


this  principle  was  quite  within  the  reach  of  Archimedes,  but 
was  not  established  until  nearly  twenty  centuries  after  his 
time. 

As  an  inevitable  corollary  of  Varignon's  proof  of  the  prin- 
ciple of  moments  we  arrive  at  the  mechanical  rule  that  in  all 
cases  of  the  parallelogram  of  forces  and  in  all  cases  of  statical 
equilibrium  of  forces  in  a  plane  the  algebraic  sum  of  the 
moments  of  the  forces  must  be  zero. 

Having  established  the  principle  of  moments,  Varignon  then 
applies  it  to  many  examples  of  equilibrium  in  rigid  bodies  and 
in  machines.  He  used  it  for  the  solution  of  all  problems  of  all 
the  simple  machines  and  founded  a  whole  system  of  statics 
on  this  idea  of  balanced  moments. 

For  the  exposition  of  the  simple  machines  it  is  perhaps  easier 
for  the  student  to  grasp  than  the  principle  of  virtual  velocities 
which  was  established  earlier. 


THE   MODERN    PERIOD.  8 1 

Varignon  in  his  Mecanique  refers  all  cases  of  equilibrium 
back  to  his  proof  of  moments  as  a  criterion  and  presents 
therefore  a  harmonious  theory  of  mechanics.  His  exposition 
is  essentially  geometrical  and  graphical,  and  is  based  upon 
Stevinus'  triangle  of  forces  the  proof  of  which  Varignon  states 
as  axiomatic  in  Lemma  II  of  his  book.  His  method  of  pre- 
sentation and  his  proofs  are  a  great  advance  over  those  of 
Archimedes  and  Stevinus. 

Many  of  the  modern  text-book  methods  in  statics  are 
copied  directly  from  this  Mecanique  and  used  verbatim  to 
this  day  in  class-rooms.  The  method  of  the  parallelogram 
of  velocities  and  of  forces  and  the  method  of  moments  as 
applied  in  the  simple  machines  are  in  daily  use  among  engineers. 

Algebra  and  geometry  deal  with  fixed  quantities,  but  with 
the  development  of  dynamics,  mathematics  was  called  upon 
to  investigate  and  express  quantities  whose  value  is  continually 
changing.  In  the  latter  half  of  the  seventeenth  century,  this 
need  was  met  by  the  invention  of  that  branch  of  mathematics, 
called  the  calculus.  As  has  been  noted  Newton  invented  one 
method  of  studying  the  relative  changes  in  dependent  quanti- 
ties by  considering  the  ratio  of  change  of  their  elements,  which 
method  of  studying  "flowing  quantities"  he  called  fluxions. 

At  about  the  same  time  Leibnitz,  feeling  the  need  of  some  such 
method,  developed  his  system  of  studying  change  by  infinitely 
small  differences  or  by  the  "method  of  infinitesimals."  The  fun- 
damental idea  and  the  purpose  of  the  two  systems  is  much  the 
same.  Each  calculus  consists  of  two  branches:  (i)  differential 
calculus  which  comprises  methods  of  deducing  the  relations 
between  infinitely  small  differences  of  quantities  from  the 
relations  of  the  quantities  themselves;  (2)  the  integral  calculus 
which  treats  of  the  inverse  process  of  determining  the  relations 
of  the  quantities  themselves  when  the  relations  of  their  in- 
finitely small  differences  is  known. 

Newton's  theory  of  flux  or  flow  was  better  suited  to  Me- 
chanics than  Leibnitz's  concept  of  instantaneous  changes  but 
the  latter's  notation  was  found  more  serviceable  and  has 
generally  displaced  Newton's  symbols.  It  was  perhaps  a 
hundred  years  before  this  method  was  generally  accepted, 
6 


82  THE   SCIENCE   OF  MECHANICS. 

recognized  and  developed.  When  we  consider  that  all  nature 
varies  continually,  the  importance  of  this  mathematical  method 
of  treating  variables  is  obvious.  With  this  instrument  once 
mastered  by  the  investigators,  advance  in  Mechanics  became 
rapid. 

The  transactions  of  the  learned  societies  of  this  period  are 
filled  with  discussions  as  to  the  integrity  of  the  calculus 
methods,  and  with  numerous  isolated  memoirs  on  its  utility  in 
one  problem  or  another  in  mechanics.  Among  the  earliest  and 
most  noteworthy  of  these  is  perhaps  that  on  page  22,  "Memoirs 
de  I'Academie  des  Sciences  de  Paris,  1700,"  in  which  Varignon 
clearly  presents  for  the  first  time,  the  differential  equations  of 
motion : 

dx  dv 

dt   ^'"'        Jt^^' 

d^x  dv       . 

df   "•''      ^dx^''' 

These  equations  express  completely  the  circumstances  of 
rectilinear  motion  for  every  condition  of  acceleration  or  retar- 
dation.  Newton  had  stated  these  laws  geometrically  (Principia, 
Lib.  I,  sect.  7;  Lib.  II,  sect,  i)  but  Varignon  seems  to  have 
been  the  first  to  advocate  their  expression  in  the  notation  of 

,  the  calculus.  He  aspired  to  free  Dynamics  from  the  encum- 
brance of  purely  geometrical  proofs  by  using  this  lately  invented 

)  method  of  the  calculus,  and  showed  how  acceleration  might 

J  be  expressed  by  the  calculus,  thus  helping  to  clear  the  way  for 

I  an  analytical  mechanics. 

'^  Like  his  predecessors  he  was  greatly  interested  in  hydro- 
statics and  hydraulics  and  is  to  be  credited  with  the  earliest 
clear  proof  of  the  important  principle  that  the  velocity  of 
efflux  of  a  liquid  is  equal  to  ^2gh. 

Starting  with  the  relation  between  force  and  the  momentum 
Ft  =  mv  and  denoting  the  area  of  the  orifice  by  a,  the  head  of 
liquid  by  h,  specific  gravity  by  S,  acceleration  due  to  gravity 
by  g,  the  velocity  of  effiux  by  v,  and  by  7"  a  small  interval  of 
time  we  have, 


THE  MODERN    PERIOD. 


83 


avTS 
ahS.T= V, 


gh  =  v^. 

In  the  formula  ahS  represents  the  pressure  acting  during 
the  time  T  on  the  mass  of  Hquid  avTS/g.  But  since  z;  is  a 
final  velocity  we  get  more  exactly, 

V 

a-TS 
ahS.T  =  ■ -v 


or 


v^  =  2gh     or,     V  =  ^2gh. 

Varignon's  contributions  may  be  summed  up  then  as: 

1 .  Proof  of  the  principle  of  moments. 

2.  A  complete  system  of  statics  based  on  moments. 


3.  The  differential  equations 


dx 
dt 

dv 
dt 

d^x 
df 


=  V, 


=  a, 


=  a. 


4.  The  equation  for  velocity  of  efflux  in  hydraulics,  v^  =  2gh. 
That  these  are  masterly  contributions  to  the  science  of 
mechanics  is  self-evident. 

REFERENCES. 

Nouvelle  Mecanique,  ouvrage  posthume  de  M.  Varignon,  2  vols.,  1725. 
Memoirs  d'l  Academie  des  Sciences  de  Paris. 
Geschichte  der  Principien  der  Mechanik,  Diihring. 
Mach's  Science  of  Mechanics. 

Leibnitz,  the  Bernoullis  and  Euler. 
Two  hundred  years  ago  the  facilities  for  the  spread  of 
scientific  progress  and  invention  were  meagre,  and  in  general 


84  THE   SCIENCE   OF  MECHANICS. 

unless  the  studies  of  the  philosophers  developed  something 
that  bore  directly  upon  some  immediate  practical  problem  of 
the  time,  no  general  notice  was  taken  of  their  work.  However 
after  the  principles  of  Statics  and  Dynamics  which  the  re- 
searches of  Stevinus,  Galileo,  Huygens,  Newton  and  Varignon 
had  developed,  came  to  be  understood  among  scholars,  the 
custom  of  sending  out  challenge  problems  arose. 

These  competitions  developed  many  small  points  which  in 
the  aggregate  amounted  to  a  very  considerable  contribution. 
Slowly  a  universal  method  of  mechanical  reasoning  and  nota- 
tion came  into  use,  which  was  understood  in  Florence  and  Paris 
^as  well  as  in  Berlin,  London  and  St.  Petersburg.     It  consisted 


\ 


in  the  reduction  of  questions  concerning  force  and  motion  to 
problems  in  pure  geometry  and  calculus. 

This  method  which  began  with  the  crude  picture  diagrams 
of  Stevinus  grew  into  the  formal  abstract  geometrical  me- 
chanics of  Newton's  "Principia"  and  by  the  genius  of  Varignon 
and  others  was  then  expressed  analytically.  Once  the  analyt- 
ical method  was  developed  it  became  the  custom  for  the 
Philosophical  Societies  of  Paris,  London,  Berlin  and  St.  Peters- 
burg to  offer  prizes  for  solutions  to  various  problems  in  me- 
chanics, and  there  resulted  a  period  of  great  activity  in  the 
application  and  extension  of  the  fundamental  contributions  of 
Stevinus,  Galileo,  Huygens,  Varignon  and  Newton. 

Thus  the  calculus  of  Newton  and  Leibnitz  came  to  be  applied 
in  a  great  variety  of  problems  in  mechanics,  and  ultimately, 
this  method  displaced  entirely  the  geometrical  method.  Dur- 
ing this  period  both  methods  were  often  used.  Numerous 
mechanical  problems  were  proven  by  both  methods  separately 
and  much  was  achieved  in  a  disjointed  unconnected  way. 

Most  active  among  those  who  took  part  in  this  development 
were  Gottfried  Wilhelm  von  Leibnitz  (1646-1716),  Leonhard 
Euler  (1707-1783),  and  the  Bernoullis — James  (1654-1705), 
John  (1667-1748)  and  Daniel  (1700-1782). 

Leibnitz,  through  his  papers  in  the  "Acta  Eruditorum," 
which  he  founded,  familiarized  continental  writers  with  his 
powerful  method  of  analysis.  Though  Newton  probably  de- 
veloped the  calculus  earlier  by  the  method  of  fluxions,  he  had 


THE   MODERN   PERIOD.  85 

presented  his  Principia  on  the  geometrical  method  and  the 
English  for  a  long  time  followed  the  geometrical  method.  It 
was  not  till  181 7  that  the  differential  calculus  was  introduced 
into  the  curriculum  at  Cambridge  and  came  into  general  use 
in  England.  On  the  continent  however,  the  method  of  Leib- 
nitz came  into  use  almost  immediately  and  generally.  Leib- 
nitz and  James  Bernoulli,  his  brother  John,  the  latter's  sons 
Nicolas  and  Daniel,  and  their  friend  Euler  were  most  active 
in  this  work  of  applying  the  Leibnitzian  analysis  to  various 
problems  in  mechanics. 

This  was  a  period  of  development  during  which  there  was 
often  acrimonious  controversy.  Two  men  sometimes  arrived 
at  similar  or  very  similar  results  by  different  routes  and  then 
entered  into  a  wordy  conflict  over  their  methods,  both  of 
which  often  proved  to  be  correct.  But  great  and  lasting  good 
came  of  all  these  discussions  for  they  served  to  clear  up  and 
define  the  fundamental  concepts  of  the  science  and  develop 
methods  and  forms  of  proof  which  later  masters  correlated 
into  formal  treatises. 

Leibnitz  does  not  appear  to  have  had  a  perfectly  clear  con- 
ception of  mass.  He  speaks  of  a  body  as  "corpus"  and  of  a 
load  or  weight  as  "moles"  and  only  once  does  "massa"  occur. 
He  makes  the  distinction  however  between  "vis  mortua"  (pres- 
sure) and  "vis  viva"  (moving  force).  His  ideas  of  the  measure 
of  force  also  were  not  clear.  He  noticed  that  in  machines  in 
equilibrium  the  loads  are  inversely  proportional  to  the  veloci- 
ties of  displacement,  so  he  measured  the  force  by  the  product 
of  the  body  ("corpus"  or  "moles"),  into  velocity.  So  far  he 
was  in  accord  with  the  notion  of  Newton  and  Descartes  who 
regarded  momentum  as  the  measure  of  force,  but  Leibnitz 
held  that  such  measure  of  force  is  only  accidental  and  that  the 
true  measure  of  force  is  determined  by  the  method  of  Galileo 
and  Huygens,  viz:  by  the  mass  times  the  velocity  squared. 

In  1686  in  the  "Acta  Eruditorum,"  Leibnitz  attacked  Des- 
cartes' conception  under  the  title  "A  short  Demonstration  of 
a  Remarkable  Error  of  Descartes  and  Others  concerning  the 
Natural  Law  by  which  they  think  the  Creator  always  preserves 
the  same  Quantity  of  Motion ;  by  which  however,  the  science 


86  THE   SCIENCE  OF  MECHANICS. 

of  mechanics  is  totally  perverted."  This  dispute  continued 
to  agitate  philosophers,  until  the  constancy  of  Smy  and  2mz^^ 
was  realized.  Certainly  Leibnitz  was  not  perfectly  clear  upon 
this.  His  discussion,  however,  helped  toward  a  solution  of 
the  difficulty. 

James  Bernoulli,  a  friend  and  admirer  of  Leibnitz,  applied 
the  calculus  to  various  problems  in  mechanics  with  marked 
success.  He  was  the  first  to  work  out  a  formula  for  the 
isochronous  curve  and  for  the  catenary  curve.^ 

Galileo  had  assumed  that  a  uniform  flexible  string  supported 
at  its  two  extremities  and  acted  upon  by  gravity  would  hang 
in  a  parabolic  curve.  James  Bernoulli  successfully  applied 
analysis  to  this  problem  of  the  "chainette"  as  he  called  it, 
and  to  derive  the  formula  for  the  catenary  curve.  Having 
solved  it  himself  he  proposed  it  as  a  challenge  problem  in  1691. 
The  four  mathematicians  who  solved  it  successfully  were  Leib- 
nitz, Huygens  and  James  and  John  Bernouilli.  Their  solu- 
tions appear  in  the  Acta  Eruditorum  for  1691,  pages  273-282, 
and  also  in  the  Philosophical  Transactions  of  the  Royal  Society 
at  London,  1697. 

From  the  physical  point  of  view  it  is  easily  seen  that  equi- 
librium exists  when  all  the  links  of  the  chain  have  sunk  as 
low  as  possible,  so  that  no  link  can  sink  lower  without  raising 
part  of  the  chain  of  equal  mass  higher,  in  consequence  of  the 
connections.  This  state  of  equilibrium  exists  when  the  center 
of  gravity  has  sunk  as  low  as  it  can  sink.  The  mathematical 
problem  then  resolves  itself  into  the  problem  of  determining 
the  curve  that  has  the  lowest  center  of  gravity  for  a  given 
length  between  A  and  B.  The  equation  of  such  a  curve 
Bernoulli  determined  with  the  aid  of  the  calculus. 

James  Bernoulli  also  extended  the  method  of  analysis  to  the 
study  of  the  curve  of  an  elastic  rod  with  a  weight  fixed  at  the 
end.  This  he  put  forth  under  the  title  of  "Elastica."  He  also 
discussed  the  problem  of  the  flexible  sheet  or  impervious  sail 
filled  with  a  liquid  which  he  presented  under  the  titles  "line- 
taria"  and  "volaria,"  and  investigated  the  theory  of  cycloidal 
lines  and  various  spiral  and  logarithmic  curves.     His  writings 

^"Opera,"  Tom.  I,  p.  449. 


THE  MODERN   PERIOD.  87 

include  an  edition  of  Algebraic  Geometry  and  the  "Ars  Con- 
jectandi,"  in  which  he  established  the  fundamental  notions 
of  the  calculus  of  probabilities. 

Johann  Bernoulli,  brother  of  James,  was  the  most  prominent 
and  successful  professor  of  mathematics  of  his  time,  holding 
the  chair  at  Basel  and  Groningen.  His  lectures  contain  the 
earliest  use  of  the  term  "integral"  and  show  the  first  effort  to 
construct  an  integral  calculus  as  a  set  of  general  rules  or  a 
body  of  mathematics.  Before  this,  investigators  had  treated 
each  problem  of  integration  by  itself.  He  made  himself  a 
master  of  the  calculus  and  applied  it  with  marked  success  to 
many  problems. 

He  was  the  author  of  the  famous  challenge  problem  of  the 
"brachistochrone"  which  he  propounded,  1697. 

In  abbreviated  form  it  is  as  follows: 

"Acutissimis  qui  toto  orbe  florent  Mathematicis" 

Johannes  Bernoulli,  Math.  P.  P. 

"Problema  Mechanico-Geometricum  de  linea  celerrimi  de- 
scensus 

"Determinare  lineam  curvam  data  duo  puncta,  in  diversis 
ab  horizonte  distantiis,  et  non  in  eadem  recta  verticali  posita, 
connectentem,  super  qua  mobile,  propria  gravitate  decurrens 
et  a  superiori  puncto  moveri  incipiens,  citissime  descendat  ad 
punctum  inferius 

Groningse  ipsis  Gal.  Jan.  1697." 
Which  may  be  freely  translated : 
A  Challenge  to  the  Keenest  Mathematicians  of  the  World, 

From  John  Bernoulli,  Prof,  of  Mathematics. 
A  Mechanico-Geometrical  Problem  of  the  Gurve  of  Swiftest 
Descent.  

Find  the  curve  of  quickest  descent  between  any  two  given 
points  at  different  distances  from  a  horizontal  line  and  not  in 
the  same  vertical  straight  line. 

Groningen,  January  i,  1697. 


88  THE   SCIENCE   OF  MECHANICS. 

The  correct  solutions  were  given  by  Leibnitz  (Acta  Erudit., 
1697,  p.  203),  by  Newton  (Phil.  Trans.,  1697,  No.  224,  p.  389), 
by  L'Hopital  (Acta  Erudit.,  1697,  p.  217)  and  John  Bernoulli 
(Acta  Erudit.,  1697,  p.  207). 

Bernoulli's  methods  are  the  methods  of  the  present  time  as 
the  following  quotations  from  his  "Lectiones  Mathematice, 
38,  39,  40,  Opera,  Tom.  Ill,"  will  show: 

A  flexible  string  fixed  at  any  two  points  A  and  B  is  acted 
upon  by  gravity.  If  we  suppose  the  mass  of  the  string  to  vary 
according  to  any  assigned  law  as  we  pass  from  one  point  to 
another,  to  find  the  equation  of  the  catenary  of  rest.  Con- 
versely the  curve  being  known,  to  determine  the  law  of  mass 
of  the  string. 

Let  the  axis  of  y  extend  vertically  upwards,  and  the  axis  of 
X  be  horizontal,  the  plane  xOy  coinciding  with  the  plane  which 
contains  the  catenary.     Then  since, 

X  =  o,     y  =  -  g, 

we  have,  by  previous  equations,  section  I, 

d   I   dx\  ,  ^ 

d  /   dy\ 

JsVTs)  = '^^'  (^) 

Integrating  the  equation  (a)  we  get 

dx 
ds 

Where  C  is  a  constant  quantity. 

Let  T  denote  the  tension  at  the  lowest  point,  then  evidently 
T  =  C,  and  therefore 

dx 


'5-.=  ^-  W 


From  {h)  and  (c),  we  have 


d  dy 
dsdx 


THE  MODERN   PERIOD.  89 

And  therefore, 


dy 
dx 


=  I  mgds,  (d) 


but  at  the  lowest  point  of  the  catenary  dy/dx  =  o  and  there- 
fore, supposing  a  to  be  the  value  of  S,  at  the  lowest  point, 


dy 
dx 


I    mds.  (e) 


If  m  be  given  in  terms  of  the  variable  x,  y,  s  the  form  of  the 
catenary  may  be  determined  from  (d). 
Again,  differentiating  (d)  we  obtain 


a  formula  by  which  m  may  be  computed  for  every  point  of 
the  string  when  the  form  of  the  catenary  is  given.  Also  from 
(c)  we  get 

which  gives  the  tension  at  any  point  of  the  catenary  when  its 
form  is  known. 

Another  example  is  that  on  page  497,  Tom.  Ill,  Lectiones 
Mathematice;  Opera. 

A  flexible  string  AOB  fixed  at  two  points  A  and  B  is  acted 
upon  by  gravity,  the  mass  at  any  P  varies  inversely  as  the 
square  root  of  the  length  OP  measured  from  the  lowest  point 
0;  to  find  the  equation  of  the  catenary. 

Let  the  origin  of  co-ordinates  be  taken  at  0,  x  being  hori- 
zontal, and  y  vertical,  and  the  plane  of  xy  coinciding  with  the 
plane  of  the  catenary,  also  let  0  be  the  origin  of  5. 

Then,  if  /x  be  the  mass  at  end  of  a  length  C  from  the  lowest 
point, 


90  THE   SCIENCE   OF  MECHANICS. 

and  therefore  i,  d,  a  being  in  the  present  case  zero,  we  have 

Jo     ^'^ 

hence  putting  for  sake  of  brevity 


dy  _ 


2gnC^  I 


we  get 


dy  _  {S\^dy'^      S 
dx  ~  [jsJ  ^2  =  "^ » 

^d^df  ^ds  ^/  df\i^ 

dxdx^      dx       \  dx^/ 


d_dy^ 
dxdx^ 

7  =  1; 


{'+%) 


integrating  with  respect  to  x  we  obtain, 


^^(■+f)'  =  '=  +  ^' 


but   X  =o,   dy/dx  =  o   simultaneously;    hence    C  =  2B    and 
therefore 

squaring  and  transposing 
df 


dy'       I  \2 


2^dy  =  {{x  +  2^y  -  4/32}^(fx; 
integrating  we  have 

C  +  2^y  =  l{x-\-  2/3)(3c2  +4^^)1-2/32  log{x  +  2/3  +  (x2  +  ^^xf] . 
But  X  =  o,  3'  =  o,  simultaneously;  hence 

C  =  2^2  log  (2^), 


THE   MODERN   PERIOD.  9I 

hence,  eliminating  C, 

2Py  =  i{x  +  2^){x-  +  4^:.)^  -  2^  log"  +  ^^  +  f f  +  ^^^^^  , 

which  is  the  required  equation  of  the  catenary. 
Cor.     From  (a)  we  get 

ds       X  +  2/3 
dx  2/3 

and  therefore,  by  (i,/) 

^ds        T  ^ 
'=^dx=T^^'+'^^^ 

which  gives  the  tension  at  any  point  of  the  curve. 

On  page  502,  of  Tom.  Ill,  Opera,  we  find  the  interesting 
problem : 

To  find  the  law  of  variation  of  the  mass  of  a  catenary  acted 
upon  by  gravity  so  that  it  may  hang  in  the  form  of  a  semi- 
circle with  its  diameter  horizontal. 

The  notation  remains  the  same  as  in  the  preceding  problem, 
and  the  equation  of  the  catenary  is 

x^  =  2ay  —  y^j 

where  a  denotes  the  radius  of  the  semi-circle;  hence 

a^  —  x'^  =  {a  —  yY, 

y  =  a  —  (a^  —  x-)^, 

dy  X  d'^y  a^ 


also 


dx      {a^  —  x^)^       dx^      (a^  —  x"^)^  * 

ds^  dy"^  a^  ds  a 

dx"^  dx"^  ~  a^  —  x^ '     dx      (a^  —  x^)^* 


and  therefore  by  (i,  e) 

.d^y 


T 
dx^  Ta  Ta 

m  = 


ds       ga^  —  x"^      g(a  —  yY 
dx 


92  THE   SCIENCE  OF  MECHANICS. 

or  the  mass  at  any  point  varies  inversely  as  the  square  of  the 
depth  below  the  horizontal  diameter  of  the  circle.  Cor.  By 
(i,/)  we  have  for  tension  at  any  point 

ds  Ta  Ta 


dx      {a^  —  x^)*      a—  y^ 

These  proofs  show  great  facility  in  handling  the  calculus 
but  they  are  an  extension  of  known  ideas  rather  than  a  new 
contribution.  Nevertheless  the  methods  of  Johann  Bernoulli 
exerted  a  great  influence  upon  the  development  of  the  science 
in  extending  the  mathematical  or  analytical  method  of  treat- 
ment. 

As  to  his  new  contributions,  he  set  forth  the  principle  of 
virtual  velocities  in  a  letter  to  Varignon,  introduced  the  symbol 
g,  and  assisted  him  in  arriving  at  the  formula  v^  =  2gh,  which 
had  been  previously  stated, 

v^  :  vi^  ::  hi  :  h. 

This  Bernoulli  was  a  profound  scholar  and  wrote  on  a  great 

variety  of  topics  as  will  be  seen  from  the  following  selection 

of  titles  from  his  Opera  Omnia  published  at  Lausanne  in  1742. 

I.     Dissertatio  de  Effervescentia  and  Fermenta- 

tione. 
II.     Novum  Theorema  pro  Doctrina  Conicarum. 

III.  Inventio  curvse  geometricae  quae  spirali  aequa- 

tione. 

IV.  Solutio  Problematis  Funicularii. 

V.     Curvae  sui  evolutione  se  ipsas  describentes. 
XVIII.     Dissertatio  physico  anatomica  de  motu  mus- 
culorum. 
LIII.     Disputatio  medico  physica  de  nutritione. 
XC.     De  motu  pendulorum  et  projectilium. 
XCIX.     Demonstratio  principii  Hydraulici  de  veloci- 
tate  per  foramen  et  vase  erumpentis. 
CXL.     Meditationes  de  Chordis  vibrantibus. 
XXIV.     Cycloidis  evoluta  ipsa  est  cyclois. 
XXXIII.     Varia  Problemata  Physico-Mechanica. 


THE   MODERN   PERIOD.  93 

The  most  interesting  of  these  is  the  first  part  of  the  third 
volume,  ;"Discours  sur  les  Loix  de  la  communication  du  mouve- 
ment,  contenant  la  solution  de  la  premiere  Question  proposee 
par  Messieurs  de  Tacad^mie  Royale  des  Sciences  pour  I'annee 
1724." 

As  a  preface  to  it  Bernoulli  says:  "The  author  of  this  dis- 
course has  the  honor  to  present  it  to  the  Academy.  It  was 
composed  on  the  occasion  of  the  first  of  the  questions  pro- 
pounded by  the  Society  to  the  savants  of  Europe.  Messrs. 
Huygens,  Mariotte,  Wren,  Wallis  and  various  other  mathe- 
maticians have  written  worthily  on  this  subject  and  given 
us  rules  for  impulse.  But  not  satisfied  with  taking  a  general 
rule  for  the  most  simple  cases,  by  a  kind  of  induction,  the 
author  has  followed  a  method  different  from  theirs  and  more 
natural. 

"He  goes  back  to  the  sources  and  taking  up  all  that  is 
known  of  the  subject,  it  is  on  principles  of  mechanics  that 
he  deduces  like  corollaries  particular  rules  for  each  case.  Up 
to  this  time  we  have  had  only  a  confused  idea  of  the  force  of 
bodies  in  motion  to  which  M.  Liebnitz  has  given  the  name 
vis  viva.  The  author  has  not  only  attempted  to  bring  the 
discussion  down  to  date  and  to  explain  the  difficulty  between 
Leibnitz  and  those  of  the  opposite  party,  he  has  attempted  to 
prove  by  demonstrations  direct  and  entirely  new,  a  truth  which 
M.  Leibnitz  never  proved  except  indirectly. 

"He  proposes  to  show  that  the  vis  viva  of  a  body  is  not 
proportional  to  its  simple  velocity  as  commonly  believed  but 
to  the  square  of  the  velocity  and  he  hopes  to  prove  what  he 
shall  say,  so  that  no  one  shall  any  longer  doubt  the  truth  of  this 
proposition:  moreover,  he  proposes  to  determine  that  which 
results  from  the  shock  of  a  body  which  encounters  two  or 
several  others  following  different  directions,  a  problem  so  diffi- 
cult that  no  one  has  yet  solved  it.  And  indeed,  how  could 
any  one,  since  its  solution  presupposes  an  exact  comprehension 
of  the  theory  of  vis  viva? 

"This  theory  opens  an  easy  way  to  several  important 
truths.  It  has  given  the  author  a  solution  of  the  preced- 
ing problem  which  seems  somewhat  peculiar  and  a  method 


94  THE   SCIENCE  OF  MECHANICS. 

of  determining  the  actual  loss  of  velocity  in  a  resisting 
medium  and  an  easy  way  of  finding  the  center  of  oscillation 
in  compound  pendulums."  He  then  goes  on  to  expound  the 
principle  of  virtual  velocities  and  the  notion  of  energy  as 
measured  by  the  mass  and  the  velocity  squared. 

Talent  for  mathematics  seems  to  have  run  in  the  Bernoulli 
family.  Several  of  the  younger  generations  were  famed  for 
their  writings  and  teaching,  among  them,  the  three  sons  of 
John,  viz:  Nicholas,  Daniel  and  John,  Jr.,  and  the  two  sons  of 
John,  Jr.,  John,  3rd,  and  Jacob. 

Of  these  Daniel  Bernoulli  (i  700-1 782)  was  the  most  promi- 
nent. He  was  professor  at  St.  Petersburg  and  at  Basel,  a 
famous  mathematician,  and  winner  of  the  French  Academy 
prize  ten  times.  His  chief  work  in  mechanics  was  on  hydro- 
dynamics and  the  solution  of  the  problems  of  vibrating  cords, 
in  which  we  find  ingenious  extensions  of  known  principles  of 
mechanics  by  the  aid  of  the  calculus. 

Euler  ( 1 707-1 783),  the  pupil  and  friend  of  Johann  Bernoulli 
and  friend  of  his  sons,  carried  the  integral  calculus  to  a  high 
degree  of  perfection  and  invented  numerous  solutions  of  me- 
chanical problems.  His  strength  lay  rather  in  pure  than  in 
applied  mathematics.  Euler's  principal  contributions  are  set 
forth  in  his  "Methodus  inveniendi  lineas  curvas  maximi  mini- 
mive  proprietate  gaudentes"  (1744)  in  which  he  presents  the 
method  of  co-ordinate  analysis  and  shows  the  properties  of 
maximum  and  minimum  of  various  curves. 

He  also  published  at  St.  Petersburg  in  1736  his  "Mechanica 
sive  Motus  Scientia  Analytice  Exposita"  which  is  sometimes 
referred  to  as  the  first  book  of  Analytical  Mechanics.  In 
this,  he  still  adheres  in  part  to  the  old  method  of  geometrical 
presentation  of  mechanics,  but  his  general  method  is  to  resolve 
all  forces  in  three  fixed  directions,  X,  Y,  Z.  This  makes  his 
presentations  and  computations  lucid  and  symmetrical. 

As  an  example,  note  the  method  of  the  following  discussion 
from  page  237,  Tom.  I  of  "Mechanica,"  on  tangential  and 
normal  resolution  in  curvilinear  motion. 

A  particle  is  projected  with  a  given  velocity  in  a  given 
direction,  and  is  acted  upon  by  a  constant  force  in  parallel  lines, 
to  determine  the  path  of  the  particle. 


THE   MODERN   PERIOD.  95 

Let  the  axis  of  X  be  taken  so  as  to  pass  through  the  initial 

place  of  the  particle,  and  let  the  axis  of  Y  be  taken  parallel 

to  the  constant  force  which  acts  toward  the  axis  of  X.     Let  / 

denote  the  constant  force. 

dy 
Then,  the  tangential  resolved  part  being  ~  f  IT^  ^^d  the 

dx 
normal  one  being  /  -7-  we  have  for  the  motion  of  the  particle 

dv  dy  , 

'ds=-fds^  ^'^ 

Integrating  (i) 

v''-  =  C  —  2jy. 

Let   V  be  the  initial  velocity;  then  y  being  zero  initially 
V^  =  C\  therefore 

j;2     ^      72     _     2/3,. 

Hence  substituting  this  expression  for  v^  in  (2) 


but 


I  dx 

dx^ 
d^y 
dx^ 


hence 


Integrating,  we  have 

Where  C  is  an  arbitrary  constant. 


96  THE   SCIENCE  OF  MECHANICS. 

Let  /3  be  the  angle  which  the  direction  of  projection  makes 
with  the  axis  of  x;  then 


hence 


Cii  +  tan2  ^)  =  72^ 

V'{^  +%)  =  sec^  ^  (^'  -  2/3'). 
V^~  =  F2  tan  ;8  -  2fy  sec^  jS, 

7^/3;  =  (72  tan2 ,3  -  2fy  sec^  |S)'  c^x; 

whence  by  integration, 

C  -  V{V^  tan2  ^  -  2fy  sec^  /3)*  =  /x  sec^  j8. 

But  X  =  o,  3'  =  o  simultaneously;  hence 

C  -  F2  tan  18  =  o, 
and  therefore 

F2  tan  jS  -  F  (72  tan^  ^  -  2fy  sec^  j8)^  =  /x  sec^  fi. 

Clearing  the  equation  of  radicals  and  simplifying,  we  obtain, 

/sec^jS 

y  =  tan  B.x  — 77^  x^. 

^  f-  2F2 

He  also  solves  in  a  similar  manner  various  problems  such  as: 

"A  particle  always  acted  on  by  a  force  in  parallel  lines, 
describes  a  given  curve;  to  determine  the  nature  of  the  force, 
the  velocity  and  the  direction  of  projection  being  given." 

And,  "A  particle  describes  a  given  curve  about  a  center  of 
force;  to  determine  the  motion  of  the  particle  and  the  law 
of  force." 

As  has  been  stated  the  name  Moment  of  Inertia  of  a  body 
was  given  by  Euler  to  the  sum  of  all  the  products  result- 
ing from  the  multiplication  of  each  element  of  the  mass  by 
the  square  of  the  distance  from  the  axis.  In  his  "Theoria 
Motus  Corporum  Solidorum,"  page  167,  Euler  says:  "Ratio 
hujus  denominationis  ex  similitudine  motus  progressivi  est 
desumpta:  quemadmodum  enim  in  motu  progressivo,  si  a  vi 


THE   MODERN    PERIOD.  97 

secundum  suam  directionem  sollicitante  acceleretur,  est  in- 
crementum  celeritatis  ut  vis  sollicitans  divisa  per  massam  seu 
inertiam;  ita  in  motu  gyratorio,  quoniam  loco  ipsius  vissollici- 
tantis  ejus  momentum  considerari  oportet,  eam  expressionem 
J  rHM  quai  loco  inertise  in  calculum  ingreditur,  momentum 
inerticB  appelemus,  ut  incrementum  celeritatis  angularis  simili 
modo  proportionale  fiat  momento  vis  sollicitattis  diviso  per 
momentum  inertise."^  This  very  useful  expression  used  so  com- 
monly by  engineers  was  developed  by  Euler  for  various  plane 
figures  and  for  solids  of  revolution.  His  method  for  finding  the 
moment  of  inertia  for  the  sphere,  right  cone,  cylinder  and 
other  figures  is  given  on  page  198  and  the  following  pages  of 
"Theoria  Motus  Corporum  Solidorum,"  as  follows: 

To  find  the  radius  of  gyration  of  a  homogeneous  sphere  about 
a  diameter.  Let  x,  x  -\-  dx  be  the  distances  of  the  circular 
faces  of  a  thin  circular  slice  of  a  sphere  at  right  angles  to  the 
diameter,  from  the  center  and  let  y  be  the  radius  of  the  section; 
then  p  denoting  the  density  of  the  sphere,  the  moment  of 
inertia  of  this  slice  about  the  diameter  will  be  equal  to 

lirpy^dx, 

and  therefore  the  moment  of  inertia  of  the  whole  sphere,  a 
being  its  radius,  will  be  equal  to 

^TTp  I       y^dx  =  Ittp   /       (a^  —  x'^ydx  =  -{^  irpa^. 

%} —a  k) —a 

As  the  mass  of  the  sphere  is  f  Trpa'  the  radius  of  gyration 

Similarly  on  page  203  the  radius  of  gyration  of  a  hollow 

^Translation. — The  scheme  of  this  notation  is  derived  by  analogy  with 
rectilinear  motion;  for  as  in  rectilinear  motion  if  it  be  increased  by  a  dis- 
turbing force  in  its  own  direction  the  increase  of  velocity  (acceleration)  is 
equal  to  the  disturbing  force  divided  by  the  mass  or  inertia,  thus  in  rotary 
motion,  since  in  place  of  the  disturbing  force  itself  we  must  consider  its 
moment  we  call  that  expression  fr-dM  which  comes  into  calculation  in 
place  of  inertia — the  moment  of  inertia — so  that  the  increase  of  angular 
velocity  in  a  similar  way  is  made  proportional  to  the  moment  of  the 
disturbing  force,  divided  by  the  moment  of  inertia. 


98  THE   SCIENCE  OF  MECHANICS. 

sphere  with  external  and  internal  diameters  a,  b,  is  proven  to  be 

2  a^  —  b^ 

k^  = 

5  a^  —  b^ 

Euler  systematized  and  perfected  the  mathematical  knowl- 
edge of  the  time.     Among  his  publications  are, 

Introductio  in  analysin  infinctorum 1748 

Institutiones  Calculi  differential 1755 

Institutiones  Calculi  Integral 1768 

also  a  development  of  the  Calculus  of  Variations. 

He  set  forth  the  principle  of  least  action, — though  Mau- 
pertuis  is  usually  given  the  credit  of  having  originated  the 
notion, — expressing  it  in  that  curious  blending  of  theology 
and  science  common  in  this  period,  in  this  fashion:  the  all- 
wise  Maker  would  not  make  anything  in  which  some  maximal 
and  minimal  property  is  not  shown. 

The  original  Latin  form  is  "Quum  enim  universi  fabrica  sit 
perfectissima,  ataque  a  creatore  sapientissimo  absoluta,  nihil 
omnino  is  mundo  contingit  in  quo  non  maximi  minimive  ratio 
qusepiam  eluceat;  quam  ob  rem  dubium  prorsus  est  nullum, 
quin  omnes  mundi  effectus  ex  causis  finalibus  ope  methodi 
maximorum  et  minimorum,  aeque  feliciter  determinari  quseant, 
atque  ex  ipsis  causis  eflficientibus,"  or  "For  since  the  fabric  of 
the  universe  is  most  perfect  and  finished  by  a  most  wise  crea- 
tor, nothing  occurs  in  the  world  in  which  some  plan  of  maxima 
and  minima  does  not  show  forth ;  therefore  there  is  no  doubt  at 
all  (!)  but  that  all  phenomena  of  the  world  are  equally  well  to 
be  determined  from  final  causes  by  the  method  of  maxima 
and  minima  and  from  the  same  effecting  causes." 

This  idea  was  taken  up  by  Euler  (Proc.  Berlin  Acad.,  1751) 
and  developed  into  a  theory  of  equilibrium  of  utility  by  the 
application  of  the  method  of  maxima  and  minima.  If  in  any 
system  we  cause  infinitely  small  displacements  we  produce  a 
sum  of  virtual  moments 

Pp  +  P'p'  +P"p"  +••• 

which  only  reduces  to  zero  in  the  case  of  equilibrium.     The 
sum  is  the  work  corresponding  to  the  displacements,  or  since 


THE   MODERN    PERIOD. 


99 


for  minute  displacements  it  is  itself  infinitely  small,  the  cor- 
responding element  of  work.  If  the  displacements  are  con- 
tinuously increased  till  a  finite  displacement  results,  their 
summation  is  a  finite  amount  of  work. 

Therefore  if  we  start  with  any  initial  configuration  of  the 
system  and  pass  to  any  given  final  configuration,  a  certain 
amount  of  work  will  have  to  be  done.  This  work  done  when 
a  final  configuration  or  a  configuration  of  equilibrium  or  equi- 
librium is  reached  is  a  maximum  or  a  minimum.  That  is  if 
any  system  is  carried  through  the  configuration  of  equilibrium 
the  work  done  is  previously  and  subsequently  less  or  greater 
than  at  the  configuration  of  equilibrium  itself.  For  equilib- 
rium, therefore 

Pp'  +  P'p'  +  P"p"  +  •  •  •  =0. 

From  this  Euler  deduced  the  principle  that  the  element  of 
work  or  the  differential  of  work  is  equal  to  zero  in  equilibrium; 
and  if  the  differential  of  a  function  can  be  put  equal  to  zero, 
the  function  has  generally  a  maximum  or  minimum  value. 


FH-   21. 

This  highly  ingenious  method  of  determining  the  equilibrium 
of  a  system  was  later  developed  by  others.  In  1749  Courtiv- 
ron,  in  a  paper  before  the  Paris  Academy  gave  it  the  form: 
"For  the  configuration  of  stable  or  unstable  equilibrium  at 


100  THE   SCIENCE  OF  MECHANICS. 

which  work  done  is  a  maximum  or  a  minimum,  the  vis  viva 
of  the  system  in  motion,  is  also  a  maximum  or  minimum  in  its 
transit  through  these  configurations." 

Euler  also  assisted  in  the  development  of  the  so-called  prin- 
ciple of  vis  viva.  He  showed  that  if  a  body  M  is  attracted 
to  a  fixed  center  C  according  to  a  certain  law  the  increase  in 
vis  viva  in  the  case  of  rectilinear  approach  is  calculable  from 
the  initial  and  terminal  distances  ro,  n.  But  the  increase  is 
the  same  if  M  passes  at  all  from  the  position  ro  to  ri  inde- 
pendently of  the  form  of  the  path  MN.  The  elements  of  the 
work  done  are  to  be  calculated  from  the  projections  on  the  radius 
of  the  actual  displacements  and  are  thus  ultimately  the  same. 

Euler  is  also  to  be  credited  with  the  first  general  use  of  r 
for  3.1416  +  and  the  application  of  his  methods  of  analysis 
to  hydrodynamics. 

We  may  sum  up  his  contributions  then  as  follows: 

1.  A  perfect  systematizing  of  the  calculus. 

2.  The  foundations  of  analytical  mechanics. 

3.  The  analytical   method   of  resolving  tangential   and 

normal  components  of  curvilinear  forces. 

4.  The  development  of  moment  of  inertia. 

5.  The  principle  of  least  action  or  maxima  and  minima 

in  equilibrium. 

6.  The  principle  that  the  increase  of  vis  viva  is  inde- 

pendent of  the  path. 

Much  of  the  work  of  D'Alembert  and  Lagrange  is  based 
on  the  contributions  or  methods  of  Euler,  and  perhaps  would 
not  have  been  possible  without  Euler's  work. 

REFERENCES. 

Jacobi  Bernoulli  Basilensis  Opera.     Geneva,  1744. 

Johanni  Bernoulli  Basilensis  Opera  Omnia.     1742. 

Opuscules  et  Fragments  inedits  de  Leibnitz.     Paris,  1903. 

Euler,  Mechanica  Sive  Motus,  1736. 

Euler,  Institutiones  Calculi  Differentialis. 

Euler,  Introductio  in  Analysin  Infinitorum. 

Correspondenz  von  Nicolaus  Bernoulli.     Basel  Library. 

Nouvelle  Mecanique.     Varignon.     Paris,  1725. 

Euler,  Methodus,  1744. 

Harnack,  Leibnitz's  Bedeutung  in  der  Geschichte  der  Mathematik. 

D.  Bernoulli,  Hydrodynamica. 

Cantor,  Geschichte  der  Mathematik. 


the  modern  period.  loi 

Jean-le-Rond  D'Alembert  (1717-1783).  ; 

As  a  result  of  the  labors  of  a  host  of  contributors  much  had 
now  been  evolved  in  mechanics  in  a  disjointed  way  and  from 
diverse  points  of  view.  The  prize  and  challenge  problems 
were  usually  very  special  and  did  not  tend  to  develop  a 
formal  presentation  of  the  science.  It  was  now  in  order  for 
some  one  to  verify,  consolidate  and  formulate  all  these  con- 
tributions. 

This  D'Alembert  did  in  his  "Traite  de  Dynamique"  (i743-) 
While  his  work  rests  upon  the  work  of  all  his  predecessors, 
and  while  he  is  particularly  indebted  to  Euler,  yet  his  Treatise 
possesses  distinctly  original  features.  He  shows  that  all 
problems  in  dynamics  may  be  regarded  as  problems  in  statics 
and  he  applies  in  their  solution  one  single  unifying  principle 
known  by  his  name  as  D'Alembert's  principle.  It  is  to  the 
effect  that  in  any  system  of  bodies  the  impressed  forces  are 
equivalent  to  the  effective  force. 

This  formal  presentation  of  mechanics  in  a  treatise  is  a 
memorable  event.  It  typifies  the  coming  of  age  of  the  science. 
Henceforth  it  has  a  character  and  unity  which  it  did  not  pre- 
viously possess.  This  is  due  to  the  fact  that  now  there  is  one 
general  guiding  principle,  D'Alembert's  Principle,  to  which 
all  problems  in  mechanics  can  be  referred  for  solution. 
Namely: — 

If  a  material  system  connected  together  in  any  way,  and 
subject  to  any  constraints,  be  in  motion  under  the  influ- 
ence of  any  forces,  each  point  of  the  system  has  at 
any  instant  a  certain  acceleration.  If  now  to  each  point 
an  acceleration  were  imparted  equal  and  opposite  to  its 
actual  acceleration,  the  velocities  of  all  points  of  the  system 
would  become  constant,  that  is,  each  particle  would  move 
as  if  free  and  unacted  on  by  any  force  whatever.  The  applied 
accelerations,  the  external  forces,  and  the  constraints  and 
mutual  or  internal  forces  of  the  system,  would  equilibrate 
one  another. 

In  the  "Traite  de  Dynamique"  this  idea  of  which  the 
above  is  a  condensed  translation  is  expressed  as  follows: — 


J 


102  THE   SCIENCE   OF   MECHANICS. 

"Probleme  General." 

"Soit  donne  un  systeme  de  corps  disposes  les  uns  par  rapport 
aux  autres  d'une  maniere  quelconque;  et  supposons  qu'on 
imprime  a  chacun  de  ces  corps  un  mouvement  particulier, 
qu'il  ne  puisse  suivre  a  cause  de  raction  des  autres  corps; 
trouver  le  mouvement  que  chaque  corps  doit  prendre." 

"Solution." 

"Soient  A,  B,  C,  etc.,  les  corps  qui  composent  le  systeme  et 
supposons  qu'on  leur  ait  imprime  les  mouvemens  a,  b,  c,  etc., 
qu'ils  soient  forces,  a  cause  de  leur  action  mutuelle,  de  changer 
dans  les  mouvements  a,  b,  c,  etc.  II  est  clair  qu'on  pent 
regarder  les  mouvemens  b,  c,  etc.,  comme  composes  des  mouve- 
mens b,  /3;  c,  7;  etc.;  d'ou  il  s'ensuit  que  le  mouvement  des 
corps  A,B,  C,  etc.;  entr'  eux  auroit  ete  le  meme,  si  au  lieu  de 
leur  donner  les  impulsions  a,  b,  c,  etc.,  on  leur  eut  donn6 
a-la-fois  les  doubles  impulsions  a,  a;  6,  j9;  C,  7,  etc.  Or  par  la 
supposition,  les  corps  A,  B,  C,  etc.,  out  pris  d'eux-memes 
les  mouvemens  a,  b,  c,  etc.,  done  les  mouvemens  a,  j8,  7,  etc., 
doivent  etre  tels  qu'ils  ne  derangent  rien  dans  les  mouvemens 
a,  b,  c,  etc.,  c'est  a-dire  que,  si  les  corps  n'avoient  recu  que  les 
mouvemens,  a,  j8,  7,  etc.,  ces  mouvemens  auroient  dil  se 
detruire  mutuellement  et  le  systeme  demeurer  en  repos. 

"De  la  resulte  le  principe  suivant,  pour  trouver  le  mouve- 
ment de  plusieurs  corps  qui  agissent  les  uns  sur  les  autres. 
De  Composez  les  mouvemens  a,  b,  c,  etc.,  imprimes  a  chaque 
corps,  chacun  en  deux  autres,  a,  a;  6,  /3;  c,  7;  etc.;  qui  soient 
tels,  que  si  Ton  n'etlt  imprime  aux  corps  que  les  mouvemens, 
a,  b,  c,  etc.,  ils  eussent  pu  conserver  ces  mouvemens  sans  se 
nuire  reciproquement ;  et  que  si  on  ne  leur  eilt  imprime  que 
les  mouvemens  a,  /3,  7,  etc.,  le  systeme  fut  demeure  en  repos; 
il  est  clair  que  a,  b,  c,  etc.,  seront  les  mouvemens  que  ces  corps 
prendront  en  vertu  de  leur  action.     Ce  qu'il  falloit  trouver." 

The  idea  was  not  entirely  new.  James  Bernoulli  in  a 
memoir  published  in  Acta  Eruditorum,  1686,  p.  356,  "Nar- 
ratio  Controversise  inter  Dn.  Hugenuim  et  Abbatem  Catela- 
num  agitatae  de  Centro  oscillationis,"  set  forth  the  idea  of 
reducing  the  determination  of  the  motions  of  material  systems 


THE   MODERN    PERIOD.  IO3 

to  the  solution  of  statical  problems.  It  is  a  direct  conse- 
quence of  Newton's  laws  rather  than  a  new  principle.  How- 
ever, to  D'Alembert  belongs  the  credit  of  clearly  setting  forth 
this  idea  and  of  founding  a  formal  mechanics  upon  it. 

In  algebraic  language  the  principle  is:  If  the  co-ordinates 
of  any  particle  w  of  a  material  system  be  x,  y,  z  and  the  ex- 
ternal forces  there  applied  be  X,  Y,  Z  the  system  of  forces 

d^-x  d?y  d'z 

^^-^^^7'    ^^-^^7/^'    ^^-^^^2- 


d'^x  d^y  d^x^ 


etc.,  acting  at  the  points  x,  y,  z  and  X2,  y^,  22,  etc.,  will  be  in 
equilibrium  in  virtue  of  the  constraints  and  mutual  reactions 
of  the  system. 

The  force  whose  components  are 

d^x  d^y  dH 

is  called  the  force  of  inertia  of  the  mass  m.  D'Alembert 's 
principle  states  that.  The  applied  forces  and  the  forces  of 
inertia  in  any  system  are  in  equilibrium. 

If  in  any  problem  the  work  be  o,  the  particular  case  of  the 
principle  of  virtual  displacement  results.  This  principle 
follows  therefore  as  a  special  case  of  D'Alembert's  principle. 

The  equation  of  vis  viva  also  follows  from  D'Alembert's  prin- 
ciple. The  integral  of  the  equations  of  motion  can  usually  be 
obtained  from  D'Alembert's  principle,  viz: 


=  o. 


Here  bx,  by,  8z  are  arbitrary  displacements  consistent  with 
the  conditions  of  the  problem.  When  the  equations  of  con- 
dition do  not  contain  the  time  explicitly,  dx  (the  actual  move- 
ment along  the  axis  of  x  during  an  infinitely  short  time)  is 
always  a  value  which  can  be  assigned  to  8x.  In  most  problems 
dx  is  a  possible  value  of  8x  and  the  same  holds  for  dy  and  dz 
similarly.     Therefore  if  this  be  admitted  as  a  legitimate  sub- 


104  THE   SCIENCE   OF  MECHANICS. 

stitution  as  is  usually  the  case,  if  we  write  dx,  dy,  dz  for  5rc,  by, 
bz,  D'Alembert's  equation  becomes 

(d^x  d^v  d^z      \ 

-^idx^^,dy  +  -^,dz)=  MXdx  +  Ydy  +  Zdz). 

Integrating  we  have 

This  is  the  equation  of  vis  viva. 

If  the  vis  viva  at  any  particular  /'  is  Xmv^  we  have 

2wz;2  -  Smy'^  =  2Xf{Xdx  +  Ydy  +  Zdz). 

If  there  be  no  forces  acting  on  the  system  its  vis  viva  remains 
constant.  The  equations  of  vis  viva  are  among  the  most 
important  in  dynamics.  They  are  the  foundation  of  the 
theory  of  energy. 

By  means  of  D'Alembert's  principle  the  equation  of  motion 
of  a  rigid  body  can  be  written  at  once.  We  have  only  to 
write  the  six  equations  of  equilibrium,  taking  into  account 
applied  forces  and  the  forces  of  inertia  and  we  have  at  once 

d^x         ,^  d^y         „  d^z 


(Pz  d^y ' 

If   ~^~d^ 

d^x  d^z 


I     d'z  d:'y\         ,   ^ 


/     dH  dH\ 


These  equations  express  the  moments  about  the  axes. 

The  comprehensive  character  and  broad  application  of 
D'Alembert's  principle  are  apparent;  other  principles  follow 
from  it  as  corollaries.  It  supplies  a  routine-form  of  solution 
for  problems,  in  a  masterly  fashion,  with  great  economy  of 
thought. 


THE   MODERN   PERIOD.  IO5 

In  his  study  of  equilibrium  and  motion  in  fluids,  and  in  the 
theory  of  vibrating  strings  D'Alembert  encountered  a  partial 
differential  equation  of  the  forms, 

8f  ~  8x^ ' 

which  he  finally  solved  in  1747.     The  solution  is  given  in  a 
paper  before  the  Berlin  Academy  as  follows: 

If  7—  be  denoted  by  p,  and  —  by  q,  then  du  =  pdx-{-pdL     But 
ox  ot 

8q      8p 
8t       8x 

by  the  given  equation,  therefore  pdt-{-qdx  is  also  an  exact 
differential,  denote  it  by  dv. 
Therefore 

dv  =  pdt-\-qdx. 
Hence 

du-\-dv  =  {pdx  -\-qdt)-\-  (pdt + qdx)  =  {p-\-q)  {dx  -\-dt) 
and 

du  —  dv  =  (pdx + qdt)  —  (pdt + qdx)  =  (p  —  q)  (dx — dt) . 

Thus  u  -{-V  must  be  a  function  of  x  +  /  and  ti  —  v  must  be  a 
function  of  x  —  /.     We  may  therefore  put 

tl-\-V  =  2<j)(x-\-t), 
U  —  V  =  2yp(x  —  t). 

Hence 

u  =  (l>(x-\-t)-{-\p(x  —  t) 

in  which  <}>  and  1/'  are  arbitrary  functions. 

In  1749  D'Alembert  published  the  first  analytical  solution 
of  the  precession  of  the  Equinoxes  and  of  the  rotation  of  the 
earth's  axis.  He  also  published  a  work  entitled  "Reflexions 
sur  la  Cause  Generale  des  Vents,"  1744,  and  three  volumes  on 
the  "Systemedu  Monde"  in  which  his  calculations  and  theories 
in  astronomy  are  set  forth. 

One  of  D'Alembert's  chief  claims  to  distinction,  in  addition 


I06  THE   SCIENCE  OF   MECHANICS. 

to  his  special  contributions,  is  that  he  put  Newton's  results 
into  the  form  of  the  Calculus  and  made  possible  their  study 
and  extension.  He  presented  in  his  "Traite  de  Dynamique," 
the  first  treatise  on  Analytical  Mechanics.  When  this  had 
been  done  the  way  was  prepared  for  a  complete  exhaustive 
treatment  of  the  entire  domain  of  Mechanics  by  the  An- 
alytical method.  This  was  done  by  Lagrange  within  five 
years  of  D'Alembert's  death. 

REFERENCES. 

Traite  de  Dynamique,  1743. 

Traite  de  I'Equibre  du  Mouvement  de  Fluides,  1744. 

Reflexions  sur  la  Cause  Generale  des  Vents,  1747. 

Recherches  sur  la  Precession  des  Equinoxes,  1749. 

Reserches  sur  difi^erent  Points  Importans  du  Systeme  du  Monde,  1756. 

Systeme  du  Monde,  3  vols.,  1754. 

W.  W.  R.  Ball,  History  of  Mathematics. 

Mach,  The  Science  of  Mechanics. 

Williamson,  Treatise  on  Dynamics. 

Bertrand,  D'Alembert. 

Condorcet,  Eloge  (French  Acad.,  1784). 

4.  The  Contribution  of  Lagrange  and  Laplace. 

Although  it  is  probable  that  Newton  used  his  method  of 
fluxions  or  calculus  in  arriving  at  the  ideas  set  forth  in  the 
Principia,  still  he  presented  them  in  geometrical  form.  Even 
so,  it  was  some  fifty  years  before  they  were  accepted  and  as- 
similated. The  next  big  step  in  advance  was  to  be  the  full 
and  complete  development  of  mechanics  by  the  analytical 
method  based  on  Newton's  laws. 

It  was  necessary  first  that  the  calculus  and  its  notation 
should  be  perfected  and  that  its  use  and  value  in  problems  of 
mechanics  should  come  to  be  recognized.  The  labors  of  Leib- 
nitz, the  Bernoullis  and  Euler  brought  this  to  pass.  Secondly 
it  was  necessary  that  the  co-ordinate  method  should  be  devel- 
oped. This  was  done  by  Descartes,  Euler  and  Maclaurin  and 
D'Alembert.  When  this  had  been  done  it  was  possible  to  ex- 
press the  results  of  Newton  in  the  language  of  the  calculus  and 
have  them  generally  received  and  accepted. 


the  modern  period.  io7 

Joseph  Louis  Lagrange  ( 1736-18 13). 

Comte  Lagrange,  one  of  the  greatest  masters  of  pure  and 
mixed  mathematics  that  ever  lived,  was  born  at  Turin  though 
of  French  extraction.  A  Senator  of  France,  a  Count  with 
the  Grand  Cross  of  the  Legion  of  Honor,  professor  in  the 
Artillery  School  of  Turin  and  in  the  Polytechnic  School  of 
France,  Director  of  the  Berlin  Academy  under  Frederick  the 
Great,  his  life  is  one  glorious  record  of  achievement. 

His  great  work  the  "Mecanique  Analytique"  is  analytical  as 
opposed  to  geometrical.  There  is  not  a  geometric  diagram  in 
it,  whereas  the  Principia  is  full  of  them,  page  on  page.  Writ- 
ten 100  years  after  Newton's  great  work,  it  is  a  grand  com- 
prehensive treatise  gathering  up  the  scattered  methods  and 
principles  of  the  preceding  century,  harmonizing  them  and 
setting  them  forth  in  concise  harmonious  algebraic  form. 
He  gives  a  general  method  by  which  every  mechanical  ques- 
tion of  solids,  liquids  or  gases  may  be  stated  in  a  single  alge- 
braic equation.  The  entire  mechanics  of  any  system,  even 
the  solar  system,  can  be  summed  up  in  a  few  equations  by  this 
method.  This  is  a  wonderful  labor-saving  and  thought- 
saving  device. 

It  was  his  boast  that  he  had  transformed  Mechanics,  (de- 
fined by  him  as  "a  geometry  of  four  dimensions")  into  a  branch 
of  analysis.  He  exhibited  the  mechanical  principles  of  his 
predecessors  as  simple  results  of  the  calculus,  and  introduced 
the  method  of  regarding  a  fluid  as  a  material  system  charac- 
terized by  free  mobility  of  its  molecules.  With  this  the  sep- 
aration between  the  mechanics  of  solids,  liquids  and  gases 
disappeared,  for  the  fundamental  equations  of  forces  could 
now  be  extended  to  hydraulics  and  pneumatics.  He  formu- 
lated a  universal  science  of  matter  and  motion,  deduced  from 
the  principle  of  virtual  velocities  by  the  method  of  generalized 
co-ordinates. 

Departing  from  the  method  of  D'Alembert  and  Euler,  in- 
stead of  considering  the  motion  of  each  individual  part  of  a 
material  system,  Lagrange  shows  how  to  determine  its  config- 
uration by  a  number  of  variables  corresponding  to  the  degrees 
of  freedom  of  the  system.     The  kinetic  and  potential  energies 


I08  THE   SCIENCE   OF   MECHANICS. 

of  the  system  can  be  expressed  in  terms  of  these  variables  and 
the  equations  of  motion  obtained  by  differentiation. 

He  gave  to  analytical  mechanics  a  complete  logical  per- 
fection, reducing  the  science  to  differential  equations  and 
developing  the  calculus  of  variations.  The  introduction  of 
the  "Mecanique  Analytique"  (1788)  is  so  simple  and  direct 
a  statement  of  the  author's  purpose  that  it  is  worthy  of  literal 
quotation. 

"There  are  already  several  treatises  on  Mechanics  but  the 
plan  of  this  one  is  entirely  new.  I  have  attempted  to  reduce 
the  theory  of  this  science  and  the  art  of  solving  the  problems 
connected  with  it  to  general  formulas,  whose  simple  develop- 
ment will  have  all  the  necessary  equations  for  the  solutions 
of  each  problem.  I  hope  that  the  manner  in  which  I  have 
tried  to  accomplish  my  object  will  leave  nothing  to  be  desired, 

"This  work  will  have  in  addition  another  advantage:  it  will 
collect  and  present  under  the  same  point  of  view  the  different 
principles,  so  far  found,  to  facilitate  the  solution  of  mechanical 
questions.  It  will  show  their  connection,  their  mutual  de- 
pendence, leaving  one  to  judge  of  their  accuracy  and  value." 

"iVo  diagram  will  be  found  in  the  work.  The  method  which 
I  follow  requires  neither  figures  nor  arguments  geometrical  or 
mechanical,  but  merely  algebraic  operations  arranged  in  a 
regular  and  uniform  order.  Those  who  are  fond  of  analysis 
will  anticipate  this  mechanics  with  pleasure,  and  be  pleased 
that  I  have  set  it  forth  in  this  way." 

Concerning  the  fundamental  principle  of  the  work,  he  says 
after  stating  D'Alembert's  principle: 

"But  there  is  another  manner  of  treatment  more  general  and 
more  severe  which  merits  the  attention  of  geometers.  M. 
Euler  gave  the  first  hint  of  it  at  the  end  of  his  treatise  on 
isomerism  printed  at  Lausanne,  1744,  showing  that  in  the  paths 
described  by  central  forces,  the  integral  of  velocity  by  the 
element  of  the  curve  always  is  a  maximum  or  a  minimum. 
This  property  M.  Euler  had  not  noticed  except  in  the  motion 
of  isolated  bodies. 

Since  that  time  I  have  considered  the  motion  of  bodies 
acting  upon  each  other  in  any  fashion  whatsoever,  and  there 


THE   MODERN   PERIOD.  IO9 

has  resulted  this  new  general  principle  that  the  sum  of  the 
products  of  the  masses  by  the  integral  of  velocities  multiplied 
by  the  elements  of  the  spaces  covered  is  constantly  a  maximum 
or  a  minimum.  Such  is  the  principle  to  which  I  give  here, 
although  improperly,  the  name  of  "least  action,"  and  which 
I  consider  not  as  a  metaphysical  principle,  but  as  a  simple  and 
general  result  of  the  laws  of  mechanics.  One  may  see  in 
volume  2,  "Memoirs  of  Jarin,"  the  use  I  have  made  of  it  for 
solving  several  difficult  problems  of  dynamics." 

This  principle  combined  with  that  of  the  conservation  of 
energy,  and  developed  according  to  the  rules  of  the  calculus  of 
variations,  gives  directly  all  the  necessary  equations  for  the 
solution  of  any  problem. 

He  then  proceeds  to  develop  his  "general  dynamic  formula 
for  the  motion  of  a  system  of  bodies  acted  upon  by  any  forces 
whatsoever,"  after  the  manner  briefly  indicated  here.  He  says 
if  the  forces  acting  upon  a  body  do  not  mutually  destroy  or 
equilibrate  themselves  as  in  statics,  then  the  forces  produce 
accelerations.  When  these  forces  act  freely  and  uniformly 
they  necessarily  produce  velocities  which  increase  with  the 
time.  One  may  regard  these  velocities  as  measures  of  the 
forces.  Let  us  suppose  now  that  of  every  accelerating  force 
we  know  the  velocity  that  it  is  capable  of  impressing  upon  a 
free  body  during  a  unit  time.  We  measure  accelerating  force 
by  the  velocity  it  produces  in  a  unit  time  supposing  the  body 
to  move  uniformly  for  that  time,  and  we  know  by  the  theorems 
of  Galileo  that  this  space  that  the  body  would  pass  over  is 
twice  the  distance  that  the  body  moves  under  a  constant 
accelerating  force,  such  as  gravity;  therefore  we  have  as  the 
velocity  by  which  to  measure  a  constant  force  twice  the 
distance  that  the  body  passes  over  in  a  unit  time.  We  must 
choose  our  units  accordingly. 

After  a  careful  development  of  these  notions  Lagrange  says 
let  us  now  consider  a  system  of  bodies  disposed  as  you  will  and 
acted  upon  by  any  accelerating  forces  you  please.  Let  m  be 
the  mass  of  any  of  the  bodies  regarded  as  point  and  let  it  be 
referred  to  three  co-ordinate  axes  by  the  co-ordinates  x,  y,  z 
at   any  instant  t,  then  dx/dt,  dy/dt,  dz/dt,  will  represent  the 


no  THE   SCIENCE  OF  MECHANICS. 

velocities  in  the  directions  of  the  axes,  if  the  body  is  abandoned 
to  itself  and  moves  uniformly.  But  if  by  reason  of  the  action 
of  accelerating  forces  the  velocities  take  on  during  the  instant 
t,  the  increments 

dx         dy         dz 

dt  ^       dt  '       dt 

one  may  regard  these  increments  as  new  velocities  and  dividing 
them  by  dt,  one  will  have  a  measure  of  the  accelerating  forces 
that  produce  them.  Taking  the  element  of  time  dt,  as  constant, 
the  accelerating  forces  will  be  proportional  to  d^x/df^,  dy'^/dt'^, 
dz^jdf,  and  multiplying  these  forces  by  the  mass  of  the  body 
upon  which  it  acts  we  have 

dH  d^y  d'^z 

^^'     ^J/^'     ^^' 

for  the  forces  moving  the  body  during  the  time  dt.  We  may 
regard  each  body  m  of  the  system  as  acted  upon  by  parallel 
forces,  then  the  total  force  will  be  equal  the  sum  of  these  parallel 
forces.  Employing  now  the  sign  d,  to  represent  differentials 
relative  to  the  time,  and  representing  the  variations  which 
express  the  virtual  velocity  by  5,  we  have 

d^x  d^y  dH 

^^^^'     ^^^^'     ^^^^ 

for  the  momenta  of  the  forces 

dH  d^y  d^z 

^^'     ^^'     ^^' 

and  for  the  sum  of  the  momenta 

'  d^x  d'^x         dH 


(d^x  d^x         d^z  ,  \ 

^8x+^,8y+j^dzjm. 


Now  let  P,  Q,  R,  etc.,  be  accelerating  forces  acting  upon  the 
system  and  p,  q,  r  their  distances,  then  the  differentials  bp, 
8q,  8r,  etc.,  represent  the  variations  of  the  lines  p,  q,  r  during 
the  variations  8x,  8y,  8z;  but  these  forces  P,  Q,  R,  tend  to 
shorten  the  lines,  therefore  their  virtual  velocities  should  be 


m 


THE   MODERN   PERIOD.  Ill 

written  —  Sp,  —  dq,  —  8r,  and  their  moments  —  mP8p, 
—  mQ8p,  —  mRbr  and  the  sum  of  all  these  forces  will  be 

-  -LiPbp  +  Qbg  +  Rhr  +  etc.)m. 

Therefore  the  sum  of  all  the  forces  acting  upon  the  body  will  be 

/  d^x  d^v  d^z     \ 

^  (^^^  +  ^2  53'  +  ^52)m  =  -  X(P8p  +  Q8q  =  RSr,etc.) 

or 

(d^x  d^y  d^z     \ 

dt^  ^^+^^3'+  ^^2  jw+2(P5^+<252+i?5r+etc.)w  =  o. 

"C'est  la  formule  generate  de  la  Dynamique  pour  le  mouve- 
ment  d'un  systeme  quelconque  de  corps."  This  formula  does 
not  differ  from  the  formula  given  in  his  Statics  says  Lagrange, 
except  in  the  terms 

d'^x  d^y  d^z 

^^'     ^^'     ^J^ 

which  express  the  accelerating  forces.  In  statics  where  the 
acceleration  is  o,  these  terms  drop  out.  Therefore  this  is  a 
general  formula  applying  to  statics  and  dynamics  and  to  solids 
and  fluids.  In  fact  the  distinction  between  statics  and  dy- 
namics and  solids  and  fluids  vanishes  except  for  the  difference 
in  substitution  in  the  formulas. 

Lagrange  then  applied  this  formula  to  many  problems  such 
as,  "Sur  le  mouvement  d'un  systems  de  corps  libres  regardes 
comme  des  points  et  animes  par  de  forces  d'attraction."  He 
was  the  first  to  make  extensive  use  of  the  calculus  of  variations. 
The  idea  of  this  is  present  in  Euler's  work  in  an  undeveloped 
form,  but  Lagrange  was  the  first  to  recognize  the  supreme 
importance  of  these  ideas  and  to  develop  the  method  of  varying 
arbitrary  constants  in  analysis.  He  successfully  applied  this 
method  to  the  investigation  of  periodical  and  secular  inequali- 
ties of  any  system  of  interacting  bodies.  These  methods  gave 
beautiful  solutions  of  such  intricate  problems  as  the  effect  of 
the  disturbance  produced  in  the  rotation  of  the  planets  by 
external  action  on  their  equatorial  protuberances.  He  also 
determined  the  first  maximum  and  minimum  values  for  the 
slowly    varying    planetary    eccentricities,    and    contributed 


112  THE   SCIENCE  OF  MECHANICS. 

memoirs  on  the  "Propagation  of  Sound"  on  the  "Motion  of 
Fluids,"  on  the  "Calculus  of  Variations,"  and  a  "Treatise 
on  Functions  and  Equations."  His  notes  on  the  Problem  of 
the  Three  Bodies,  on  Variations  of  the  Element  of  Planetary- 
Orbits,  on  Attractions  of  Ellipsoids,  and  on  the  Moon's  Secular 
Inequality  are  noteworthy. 

Lagrange  verified  Newton's  theory  and  developed  his  sug- 
gestions much  as  Newton  did  those  of  Galileo.  He  reduced 
the  whole  theory  of  mechanics  to  one  fundamental  formula, 
and  drew  clearly  the  line  between  physics  and  metaphysics. 
After  his  time  we  hear  no  more  such  fantastic  speculations  as 
were  set  forth  by  Descartes  and  Leibnitz. 

Duhring  in  his  "Geschicte  der  Principien  der  Mechanik," 
page  305,  sums  Lagrange's  contribution  in  these  words: 

"Die  Anwendung  eines  Fundamentalprincips,  welches  sich 
fiir  den  Calciil  eignet,  und  die  grunsatzliche  Durchfiihring 
der  analytischen  Entwicklungen  als  der  Haupt  eitfadens  fiir 
die  Verbindung  aller  Wahrheiten  der  rationellen  Mechanik 
zu  einem  einheitlichen  System, — das  sind  die  beiden  Hauptei- 
genschaften,  durch  welche  sich  die  Behandlungsart  Lagranges 
auszeichnet."  /.  e.,  The  application  of  a  fundamental  principle 
adapted  to  the  calculus  and  the  consistent  utilization  of  an- 
alysis as  his  main  guide  for  the  combination  of  all  the  truths 
of  rational  mechanics  into  a  unified  system,  these  are  the  two 
points  which  distinguish  Lagrange's  method. 

Laplace,  Simon  Pierre,  Marquis  De  (1749-1827). 

The  genius  of  Lagrange  was  at  its  best  in  generalization 
and  abstraction  and  he  brought  his  mind  to  practical  physical 
problems  with  difficulty.  It  was  not  so  with  his  contemporary 
Laplace,  who  was  gifted  with  shrewd  practical  sagacity  in 
addition  to  the  wonderful  mathematical  power  which  won 
for  him  the  title  of  the  "Newton  of  France." 

He  applied  himself  especially  to  the  great  problems  of  de- 
veloping an  analytical  exposition  of  celestial  motions  and  per- 
turbations, based  upon  the  law  of  gravitation,  and  he  spent 
his  life  in  tracing  the  consequences  of  the  law  of  gravitation 
as  applied  to  the  solar  system. 

The  solar  system  does  not  consist  of  several  bodies,  but  of 


THE   MODERN   PERIOD.  II3 

a  crowd  of  them  traveling  about  the  sun,  many  of  them  at- 
tended by  satelHtes;  thus  the  compHcation  of  attractions 
is  evident.  Again  the  motion  of  a  planet  at  any  time  de- 
pends not  merely  upon  its  relative  position  with  reference 
to  the  sun,  but  also  upon  the  position  of  the  other  planets  ^ 
and  of  its  own  satellites.  Added  to  this  is  the  difficulty  that 
no  planet  is  where  it  seems  to  be,  owing  to  the  effects  of 
atmospheric  refraction  and  of  the  finite  velocity  of  light.  The 
magnitude  of  the  task  that  Laplace  set  himself  is  appalling. 

Yet  he  produced  in  his  "Mecanique  Celeste"  a  work  in 
which  the  whole  theory  of  planetary  motions  is  investigated, 
and  which  offers  a  complete  solution  of  the  great  mechanical 
problem  presented  by  the  solar  system.  It  was  his  constant 
endeavor  to  "bring  theory  to  coincide  so  closely  with  observa- 
tion that  empirical  equations  should  no  longer  find  a  place  in 
astronomical  tables."  His  work  is  based  on  the  Principia  of 
Newton,  which  he  translates  into  the  language  of  the  calculus, 
and  carries  forward  and  completes  so  as  to  produce  a  mechan- 
ical theory  of  celestial  motions. 

The  "Mecanique  Celeste,"  in  five  volumes,  gives  a  full  an- 
alytical discussion  of  the  solar  system.  The  first  two  give 
methods  for  calculating  the  motions  of  translation  and  rotation 
of  the  planets,  determining  their  figures  and  solving  tidal 
problems.  The  third  and  fourth  volumes  contain  applications 
of  these  formulae  and  astronomical  tables.  The  fifth  volume 
is  historical.  The  work  is  a  complete  treatise  on  physical 
astronomy.  The  "Exposition  du  systeme  du  Monde"  is  the 
"Mecanique  Celeste"  in  popular  form  without  the  analysis. 
The  results  only  are  given  and  the  nebular  theory  is  pro- 
pounded. 

Laplace's  special  contributions  to  the  notation  of  mechanics 
are  the  Laplace  Coefficient  and  the  Potential  Function.  In 
the  course  of  his  work  of  investigating  the  figure  of  a  rotating 
fluid  mass,  the  stability  of  Saturn's  rings,  etc.,  he  came  upon 
expressions  for  the  attraction  of  an  ellipsoid  involving  an  in- 
tegration, which  he  could  not  solve.  He  discovered  however 
that  the  attracting  force  in  any  direction  could  be  obtained 
by  the  direct  process  of  differentiating  a  single  function.  He 
8 


114  THE   SCIENCE  OF  MECHANICS. 

was  then  able  to  translate  the  forces  of  nature  into  the 
language  of  analysis  so  that  he  could  consider  also  by  mathe- 
matical analysis  the  phenomena  of  heat,  electricity  and 
magnetism. 

The  function  V  which  was  named  the  Potential  Function 
by  Green  and  Gauss  about  1840  is  defined  as  the  sum  of  the 
masses  of  the  molecules  of  the  attracting  bodies  divided  by 
their  respective  distance  from  the  attracting  point. 

In  general  terms  m  being  the  mass,  and  r  the  distance  from 

the  attracting  point,  we  have 

^^       _     .   2Aw 

V  =  Limit , 

r 

or 

Am  =  o, 

if  p  is  the  density  of  the  body  at  the  point  x,  y,  z  and  a,  j8,  7 
the  co-ordinates  of  the  attracted  point 


v=  fff 

J  J  J    [{x  —  a 


pdxdydz 


)2  +  (3,  _  ^)2  +  (2  _  y)]'  ' 

the  limits  of  the  integration  being  determined  by  the  form  of 
the  attracting  mass.  Therefore  F  is  a  function  of  a,  /3,  7, 
that  is,  it  depends  on  the  position  of  the  point,  and  its  several 
differentials  furnish  the  components  of  the  attractive  force. 
As  the  integrations  did  not  usually  give  V  in  finite  terms, 
Laplace  introduced  (1785)  the  partial  differential  equation 

Since  known  as  Laplace's  equation.  Here  V^  is  called  the 
operator.  This  equation  forms  the  basis  of  all  Laplace's  re- 
search in  attractions  and  opened  up  the  whole  field  of  potential. 
This  equation  is  now  used  in  every  branch  of  physical  science. 
The  quantity  V^F  may  be  viewed  as  the  measure  of  the  con- 
centration of  F.  Its  value  at  any  point  indicates  the  excess 
of  the  value  of  F  at  that  point  over  its  mean  value  in  the  neigh- 
borhood of  the  point.  This  potential  function  laid  the  foun- 
dation of  the  mathematical  development  of  heat,  electricity 
and  magnetism. 


THE    MODERN   PERIOD.  115 

The  form  in  which  Laplace  first  gave  his  equation,  "Re- 
cherches  sur  I'attraction  des  Spheroides  homogenes"  in  Divers 
Savans,  v.  lo,  1873,  is,  in  the  polar  co-ordinate  form, 

where  ix  is  substituted  for  the  cos  9. 

If  two  points  in  space  are  determined  by  their  polar  co- 
ordinates r,  6,  CO  and  /,  6',  w',  and  T  be  the  reciprocal  of  the 
distance  between  them  expressed  in  these  co-ordinates,  then 


T=  {r^-  2rr'  W  +  Vi-,j?Vi-  ^'^  ^^g  (^  _  ^.)]  _,_  ^>^y, 

where  n  and  n'  represent  the  cos  B  and  cos  6' . 

If  this  expression  be  expanded  into  a  series  of  the  form 

-,  (Po  +  Pi-,  +  P2-2  +  •  •  •  P„-,i  •  •  •). 

where  Po,  Pi,  P^  are  known  as  Laplace's  coefficients  of  the  orders 
o,  I,  ...  a,  these  are  found  to  be  rational  integral  func- 
tions of  yi  and  /,  of  -v/l  —  /i^  cos  w  and  '^ i  —  /t'^  cos  w  and 
-v^i  —  /i^  sin  CO  and  Vi  —  ti'^  sin  co  or  of  the  rectangular  co- 
ordinates of  the  two  points  divided  by  their  distances  from  the 
origin.  The  general  coefficient  P„  is  of  a  dimensions  and  its 
maximum  value  Laplace  shows  to  be  unity  so  that  the  above 
series  will  converge  if  r'  is  greater  than  r.  He  proves  that  T 
satisfies  the  differential  equation 

^^'  ~^^^lin  I         d'T        dKrT) 

dix  "^  I  -  m'  *  <^co2  +  ^    dr''     ~  °' 

and  if  for  T  the  expanded  form  is  substituted  we  obtain  the 
general  differential  equation  of  which  Laplace's  coefficients 
are  particular  integrals 

.  ^dPa 

^ +  -o  •  -TT  +  «(«^  +  l)^^  =  O- 

dix  I  —  )u       aco'' 

Laplace's  theorem  of  these  functions  is  to  the  effect  that  if 


Il6  THE   SCIENCE  OF  MECHANICS. 

Expressions  that  satisfy  this  are  called  Laplace  functions. 
Y  and  Z  be  two  such  functions,  i  and  i'  being  whole  numbers 
and  not  identical  then 


J— I  Jo 


YiZidfidco  =  o. 


The  great  value  of  these  functions  in  physical  research  de- 
pends on  the  fact  that  every  function  of  the  co-ordinates  of  a 
point  on  a  sphere  can  be  expanded  in  a  series  by  Laplace's 
functions.  They  are  therefore  useful  in  mechanics  in  researches 
in  which  spheres  figure,  as  in  the  problem  of  the  figure  of  the 
earth,  the  general  theory  of  attraction,  and  in  electricity  and 
magnetism. 

Laplace  also  published  in  1812  his  "Theorie  analytique  des 
Probabilities,"  an  exhaustive  treatment  of  the  subject  of 
probability. 

It  cannot  be  said  of  Laplace  that  he  created  a  new  branch 
of  science  like  Galileo  or  Archimedes,  new  principles  or  a 
radically  new  method  like  Newton,  Leibnitz,  or  Descartes.  His 
work  was  one  of  verification  and  formulation  of  known  ideas 
into  grand  generalizations.  He  possessed  a  genius  for  tracing 
out  the  remote  consequences  of  the  great  principles  already 
developed,  and  he  brought  within  the  range  of  analysis  a  great 
number  of  physical  truths  which  it  did  not  appear  probable 
could  ever  be  brought  subject  to  laws  of  mechanics.  His  great 
contribution  was  the  invention  of  the  potential  function  in 
analysis,  which,  as  developed  by  him  and  later  by  Green,  Gauss 
and  Lord  Kelvin,  brought  fluid  motion,  heat,  electricity,  and 
magnetism  under  the  dominion  of  analytical  mechanics. 

REFERENCES. 

Mecanique  Analytique.     Paris,  1788. 

Mecanique  Analytique.     Paris,   i8ii. 

Exposition  du  Systeme  du  Monde.     Paris,  1873. 

Mecanique  Celeste.     Translated  by  Bowditch. 

Kelvin,  General  Integration  of  Laplace's  Differential  Equations  of  Tides. 

Diihring,  Geschichte  der  Principien  der  Mechanik. 

Todhunter,  Treatise  on  Laplace's  Functions. 

Mach,  The  Science  of  Mechanics. 

Williamson,  Treatise  on  Dynamics. 

Todhunter,  History  of  the  Mathematical  Theory  of  Attraction. 

Thomson  and  Tait,  Treatise  on  Natural  Philosophy. 


the  modern  period.  ii7 

5.  Recent  Contributions. 
The  Contribution  of  Louis  Poinsot  (i 777-1 859). 

The  contribution  of  Poinsot  to  the  science  of  mechanics  is 
one  of  method  rather  than  of  principle.  In  fact,  since  the 
time  of  Lagrange  and  Laplace  no  radically  new  principle  in  the 
science  of  mechanics  has  been  brought  forth,  with  the  excep- 
tion of  the  principle  of  conservation  of  matter  and  of  energy. 

Poinsot's  work  is  set  forth  in  two  volumes:  "Les  Elemens 
de  Statique"  and  "Theorie  Nouvelle  de  la  Rotation  des  Corps." 
He  follows  Newton's  method,  and  builds  the  science  on  force,  T 
mass,  and  acceleration  as  fundamental  concepts,  but  in  his 
exposition  the  notion  of  couples,  i.  e.,  pairs  of  parallel  forces 
acting  on  the  same  body  in  opposite  directions  has  a  prominent 
part.  This  idea  of  a  couple  was  now  new;  Poinsot  did  not 
originate  it.  It  follows  from  the  principle  of  moments  as  set 
forth  by  Varignon  in  1687,  but  nothing  worth  mentioning 
had  been  made  of  the  idea  till  Poinsot  based  a  system  of 
mechanics  on  it,  in  his  Elemens  de  Statique  in  1803.  Perhaps 
no  idea  in  mechanics  is  so  easily  comprehended,  so  useful  and 
so  fruitful  in  the  presentation  of  equilibrium  of  rigid  bodies. 
But  it  does  not  express  the  historical  development  of  the 
science.  Once  mechanics  had  been  developed,  it  was  easy  to 
formulate  a  system  of  mechanics  by  the  idea  of  the  couple, 
but  as  a  rational  primitive  conception,  the  idea  of  equilibrium 
established  in  this  way  does  not  appeal  to  the  mind. 

Poinsot  says,  in  the  preface  of  the  "Elemens":  "Dans  la 
solution  mathematique  des  problemes,  on  doit  regarder  un 
corps  en  equilibre  comme  s'il  etait  en  repos;  et  reciproquement, 
si  un  corps  est  en  repos,  on  sollicite  par  des  forces  quelconques, 
on  peut  lui  supposer  appliquees  telles  nouvelles  forces  qu'on 
voudra,  qui  soient  en  equilibre  d'elles-memes,  et  I'etat  du 
corps  ne  sera  point  change.  On  verra  bientot  de  nombreuses 
applications  de  cette  remarque."  One  may  regard  a  body  in 
equilibrium  as  if  at  rest,  and  one  may  regard  a  body  at  rest 
as  being  so,  because  the  forces  applied  to  it  balance  each  other. 
One  may  assume  various  other  pairs  of  forces  applied  to  the 
body  and  it  will  still  remain  at  rest.  This  idea  has  many 
useful  applications. 


) 


Il8  THE   SCIENCE   OF  MECHANICS. 

He  then  develops  the  idea  of  a  couple  and  sets  forth  a 
number  of  theorems  on  couples  from  which  he  evolves  the 
theory  of  the  simple  machines.  He  says:  "Nous  reduirons 
les  machines  simples  a  trois  principales  que  Ton  peut  considerer 
si  Ton  dans  I'ordre  suivant  en  regard  a  la  nature  de  I'obstacle 
qui  gene  le  mouvement  du  corps:  le  levier  le  tour  et  le  plan 
incline."  The  simple  machines  may  be  reduced  to  three  prin- 
ciples according  to  the  nature  of  points  considered  as  fixed, 
viz:  the  lever,  the  screw  and  the  inclined  plane. 

In  the  first,  the  obstacle  or  impediment  is  a  fixed  point;  in 
the  second,  it  is  a  straight  line;  in  the  third,  it  is  a  fixed  plane. 
From  these  he  develops  geometrical  theorems  on  the  simple 
machines. 

In  general,  Poinsot's  method  is  distinctly  his  own  develop- 
ment of  a  synthetic  mechanics,  based  on  Newton's  ideas. 
He  does  not  use  the  calculus,  but  develops  the  whole  system 
by  a  judicious  choice  of  fixed  points  and  by  the  action  of 
couples.  He  gives  a  self-contained  exposition  of  the  science 
which  is  useful  rather  as  a  practical  text-book  than  as  a  system 
for  advancing  the  science.  The  Theorie  Nouvelle  de  la  Rota- 
tion des  Corps  treats  of  the  motion  of  a  rigid  body  by  geometry 
and  shows  that  the  most  general  motion  of  such  a  body  can 
be  represented  at  any  instant  by  a  rotation  about  an  axis 
combined  with  a  motion  of  translation  parallel  to  the  axis, 
and  that  any  motion  of  a  body,  of  which  one  point  is  fixed, 
may  be  produced  by  the  rolling  of  a  cone  fixed  in  a  body  on  a 
cone  fixed  in  space.  This  enables  one  to  picture  the  motion 
of  a  rigid  body  as  clearly  as  the  motion  of  a  point.  The 
previous  treatment  of  the  motion  of  such  a  body  had  been 
analytical,  and  gave  no  mental  picture  of  the  moving  body. 

Poinsot's  exposition  of  statics  and  of  rotation  by  the  action 
of  couples  about  arbitrarily  chosen  fixed  points,  lines,  or  planes, 
is  valuable  as  offering  ready  practical  conceptions  of  mechan- 
ical action  for  every-day  use.  It  is  just  such  a  system  as 
one  would  expect  a  professor  in  a  technical  school  to  develop 
for  the  use  of  students  who  were  preparing  for  professional 
work  rather  than  for  research.  The  diagrams  demonstrate 
the  theorems  so  as  to  make  the  proof  almost  axiomatic  and 


THE   MODERN   PERIOD.  II9 

intuitive.  His  theorems  are  to  be  found  to-day  in  modern 
text-books  and  are  of  service  to  the  mechanical  and  civil 
engineer. 

Among  his  memoirs  are  contributions  on:  "Sur  la  composi- 
tion des  moments  et  des  aires."  "Sur  la  geometric  de  I'equi- 
libre  et  du  mouvement  des  Systemes."  "Sur  la  plan  invariable 
du  systeme  du  monde."  His  Mechanics  is  valuable  for  its 
ready  practical  methods,  rather  than  for  new  contributions 
to  the  science. 

The  Contributions  of  Simeon  Denis  Poisson 
(1781-1840). 

Poisson,  the  distinguished  young  contemporary  of  Laplace 
and  Lagrange,  was  their  equal  in  mathematical  analysis  and 
their  superior  in  grasp  of  physical  principles.  A  large  number 
of  memoirs,  on  a  wide  range  of  scientific  subjects,  testify  to 
his  ability.  In  some  of  these  he  corrected  errors  in  the  work 
of  Laplace  and  Lagrange. 

Poisson  applied  himself  particularly  to  mathematical 
physics.  He  explored  heat,  light,  electricity  and  magnetism 
by  analysis  and  originated  the  method  of  investigation  by 
"potential."     He  evolved  the  correct  equation  for  potential 

V^V  =  —  47rp 

in  place  of  Laplace's  equation 

V^F  =  o. 

This  equation  now  appears  in  all  branches  of  mathematical 
physics,  and,  according  to  some  writers,  it  follows  that  it 
must  so  appear  from  the  fact  that  the  operator  V^  is  a  scalar 
operator.  Indeed  it  may  be  that  this  equation  represents 
analytically  some  law  of  nature  not  yet  reduced  to  words. 

Poisson's  work,  "Traite  de  Mecanique"  (1853),  is  an  excel- 
lent exposition  of  rational  mechanics  by  the  method  of  the 
calculus  It  proceeds  logically  from  the  definitions  of  "corps," 
"masse"  and  "force,"  and  a  definition  of  Mechanics  "la  science 
qui  traite  de  I'equilibrium  et  du  mouvement  des  corps"  through 


120  .  THE   SCIENCE   OF   MECHANICS. 

statics  and  dynamics,  section  by  section.  Though  it  contains 
some  variations  in  mathematical  presentation,  it  contains  no 
new  principle. 

His  work  on  the  theory  of  Electricity  and  Magnetism  and 
his  "Theorie  Mathematique  de  la  Chaleur,"  1835,  present 
methods  by  which  nearly  all  physical  phenomena  may  be 
explained  in  terms  of  mathematical  mechanics.  With  this 
the  science  of  mechanics  approaches  its  highest  development. 
From  the  time  of  Poisson  up  to  the  present,  a  number  of 
investigators  have  worked  over  the  field  and  developed  the 
applications  of  known  principles  and  methods.  Among  them 
must  be  mentioned : 

Fourier,  Theorie  analytique  de  la  chaleur,  1822. 

Gauss,  De  figura  fluidorum  in  statu  sequilibrie,  1828. 

Poncelet,  Cours  de  mecanique,  1828. 

Belanger,  Cours  de  mecanique,  1847. 

Mobius,  Statik,  1837. 

Coriolis,  Traite  de  Mecanique,  1829. 

Grausmann,  Ausdehnungslehre,  1844. 

Hamilton,  Lectures  on  Quaternions,  1853. 

Jacobi,  Vorlesungen  iiber  Dynamik,  1866. 

Joule,  J.  P.,  Scientific  Papers,  1887. 

As  a  result  of  the  earnest  labors  of  these  and  others,  and  more 
particularly  by  the  patient  research  of  those  mentioned  below, 
the  nineteenth  century  saw  the  establishment  of  the  great 
mechanical  principle  of  conservation,  the  most  unifying  and 
fruitful  of  all  scientific  dogmas.  It  is  the  result  of  the  accu- 
mulated experience  of  many  inquirers  rather  than  the  achieve- 
ment of  any  individual. 

The  Law  of  Conservation. 

In  1775,  the  French  Academy  declined  to  consider  any 
further  devices  for  obtaining  "perpetual  motion,"  but  it  was 
not  till  one  hundred  years  later,  about  1875,  that  the  generali- 
zations known  as  the  Conservation  of  Matter  and  the  Con- 
servation of  Energy,  or  the  Law  of  Conservation  came  to  be 
generally  admitted  after  long  experiment  and  careful  study. 


THE   MODERN   PERIOD.  121 

The  principle  of  the  Conservation  of  Matter  was  established 
about  1780  by  Lavoisier,  (1743-94),  as  a  result  of  a  series  of 
experiments  with  the  chemist's  balance  which  indicated  that 
the  mass  of  a  given  quantity  of  matter  remains  constant 
regardless  of  change  of  state  or  of  chemical  combination. 

The  principle  of  conservation  of  energy  was  of  slow  growth. 
The  idea  of  conservation  in  nature  seems  to  have  been  dimly 
felt  as  far  back  as  the  time  of  Descartes  (1596-1650).  New- 
ton, also,  seems  to  have  had  an  idea  of  it,  though  his  de- 
velopment of  mechanics  by  the  concepts  of  work,  force  and 
distance,  blinded  him  to  the  appreciation  of  the  measure  of 
activity  by  energy.  Still  in  the  scholium  to  his  third  law,  we 
read :  "If  the  action  of  an  agent  be  measured  by  the  product  of 
the  force  into  its  velocity,  and  if  similarly  the  reaction  of  the 
resistance  be  measured  by  the  velocities  of  its  several  parts 
multiplied  into  their  several  forces,  whether  they  arise  from 
friction,  cohesion,  weight  or  acceleration,  action  and  reaction 
in  all  combination  of  machines  will  be  equal  and  opposite." 
It  is  probable  that  the  popularity  of  the  Newtonian  exposition 
of  mechanics  from  the  point  of  view  of  force  and  work,  had  a 
tendency  to  delay  the  establishment  of  this  principle  of  con- 
servation. The  concept  of  Energy  was  foreign  to  Newton's 
mechanics. 

The  principle  was  rather  a  slow  development  of  the  Huy- 
genian  idea  of  energy  and  it  came  to  the  fore,  with  the  recog- 
nition of  a  relation  between  mechanical  energy  and  heat.  The 
idea  that  heat  is  a  form  of  energy  for  which  there  is  an  exact 
mechanical  equivalent  was  first  suggested  about  1798,  by  the 
experiments  of  Count  Rumford  on  the  heat  resulting  from 
the  boring  of  cannon  and  by  the  experiments  the  following 
year,  of  Sir  Humphrey  Davy  on  melting  ice  by  friction.  This 
conception  was  at  variance  with  the  generally  held  hypothesis 
that  heat  was  of  the  nature  of  a  material  fluid. 

The  idea  languished  till  1842,  when  Julius  Robert  Mayer 
began  experimental  research  on  the  subject.  Choosing  as  the 
unit  of  heat,  the  quantity  necessary  to  raise  one  gram  of  water 
ato°  C,  one  degree  centigrade,  commonly  called  a  "calorie," 
and  for  the  unit  of  work,  one  gram  lifted  one  meter  or  a 


122  THE   SCIENCE   OF  MECHANICS. 

"gram-meter,"  the  determination  of  the  number  of  gram- 
meters  that  are  equivalent  to  a  calorie  in  energy  was  stated 
by  Mayer  as  365  from  his  experiments  on  the  heat  evolved  in 
compressing  air. 

In  1843  J.  P.  Joule  (1818-89)  undertook  the  investigation  of 
the  subject  and  invented  a  variety  of  apparatus  for  determin- 
ing the  dynamical  equivalent  of  heat  and  among  other  forms 
the  common  laboratory  method  of  descending  weights  turning 
paddle  wheels  in  a  vessel  of  water,  the  temperature  of  which 
is  determined  by  thermometers.  The  subject  now  came  up 
for  thorough  investigation  and  discussion  by  scientists.  Helm- 
holtz  maintained  the  principle  in  "Ueber  die  Erhaltung  der 
Kraft,"  1847,  and  Rankine,  Kelvin,  Clausius  and  Maxwell 
contributed  either  experimentally  or  theoretically  to  its  estab- 
lishment. 

It  is  worded  in  various  ways,  one  form  being:  In  any  system 
of  bodies  the  energy  remains  constant  during  any  reaction  or 
transformation  between  its  part.  It  is  also  stated  as:  "The 
energy  of  the  universe  is  constant." 

In  1850  Joule  obtained  his  value  423.5  gram-meters  for  the 
dynamical  equivalent  of  heat  which  for  two  decades  was  the 
accepted  value.  By  i860  research  had  verified  this  figure  by 
transformations  of  energy  through  mechanical,  electric,  mag- 
netic and  chemical  transformations  in  sufficient  number  to 
warrant  the  acceptance  of  the  principle  of  conservation  of 
energy.  Prof.  Rowland  in  1879  made  a  series  of  very  careful 
determinations  of  the  dynamical  equivalent  of  heat  using 
Joule's  stirring  or  paddle  apparatus,  and  finally  gave  the  value 
425.9  for  water  at  10°  C. 

This  principle  is,  as  Maxwell  says,  "the  one  generalized 
statement  which  is  found  to  be  consistent  with  fact,  not  in 
one  physical  science  only  but  in  all.  When  once  apprehended, 
it  furnishes  to  the  physical  inquirer  a  principle  on  which  he 
may  hang  every  known  law  relating  to  physical  actions,  and 
by  which  he  may  be  put  in  the  way  to  discover  the  relations 
of  such  actions  in  new  branches  of  science."  He  states  the 
principle  as  follows:  "The  energy  of  a  system  is  a  quantity 
which  can  neither  be  increased  nor  diminished  by  any  action 


THE   MODERN   PERIOD.  1 23 

between  the  parts  of  the  system,  though  it  may  be  transformed 
into  any  of  the  forms  of  which  energy  is  susceptible."  The 
total  energy  of  a  closed  system  is  invariable  quantity. 

Whether  the  energy  of  a  system  is  partially  in  the  kinetic 
and  partially  in  the  potential  form,  whether  the  energy  exists 
as  potential  energy  of  arrangement  of  the  gross  parts  of  a 
system,  or  as  molecular  energy,  or  electrical  energy,  or  as 
kinetic  energy  of  moving  masses,  or  of  moving  molecules,  or 
of  vibrations  of  the  ether  or  of  electrical  currents,  the  total 
quantity  of  energy  in  an  isolated  system  is  constant. 

We  have  no  acquaintance  w^ith  "absolute  energy"  or  of 
energy  apart  from  matter.  Our  knowledge  is  limited  to  energy 
changes  in  matter.  Work  done  upon  a  body  or  a  system 
increases  its  energy,  or  work  done  by  it  upon  another  body 
confers  energy  upon  it.  If  we  do  work  upon  a  body  weighing 
100  lbs.  so  as  to  raise  it  vertically  5  ft.  we  store  500  ft.  lbs.  of 
energy  in  it,  which  is  said  to  be  in  the  "potential"  form.  The 
mathematical  expression  of  energy  always  requires  two  factors. 
For  instance,  in  doing  mechanical  work  we  may  measure  the 
energy  by  the  product  of  the  force  times  the  distance,  F  XS, 
or  if  the  work  has  produced  kinetic  energy  we  measure  it  by 
the  mass  of  the  body  multiplied  by  the  square  of  the  velocity, 
i.  e.,  mv^J2.  In  case  the  mechanical  work  is  transformed  into 
heat  the  factors  become  the  specific  heat  and  the  rise  in  tem- 
perature. If  the  heating  is  produced  by  a  transformation  of 
electrical  energy,  the  electrical  energy  is  measured  by  the 
quantity  of  electricity  and  the  electromotive  force. 

From  the  principle  of  conservation  have  been  evolved  the 
three  principles  of  thermodynamics  or  of  energetics  which  are 
commonly  listed  as: 

(i)  the  conservation  of  energy; 

(2)  the  distribution  of  energy  or  the  principle  of  Carnot; 

(3)  the  law  of  least  action. 

The  second  principle  is  given  by  Clausius  in  the  form: 
"Heat  cannot  of  itself  pass  from  a  colder  body  to  a  warmer 
one."  Lord  Kelvin  put  it  thus:  "It  is  impossible,  by  means 
of  inanimate  material  agencies  to  derive  mechanical  effect 
from  any  portion  of  matter  by  cooling  it  below  the  tem- 
perature of  the  coldest  surrounding  objects." 


124  THE   SCIENCE   OF  MECHANICS. 

This  was  later  generalized  and  put  into  the  form :  The  trans- 
fer of  energy  can  only  be  effected  by  a  fall  in  tension.  This 
is  the  principle  of  Carnot  and  signifies  that  energy  always 
goes  from  the  point  where  the  tension  is  high  to  the  point 
where  it  is  low.  This  applies  not  only  to  heat  but  to  all 
known  forms  of  energy. 

If  we  imagine  a  system  of  bodies  taken  at  random  in  various 
conditions  of  temperature,  electrification,  etc.,  they  will  not 
remain  as  thrown  together,  but  a  readjustment,  with  trans- 
ferences and  transformations  of  energy  will  begin,  until  one 
of  the  factors  of  the  energy  of  all  the  bodies  has  the  same 
value  or  intensity  in  all  parts  of  the  system. 

That  is,  if  the  electromotive  force  or  the  temperature  is 
the  same  in  all  parts  of  the  system,  no  transference  takes 
place;  or,  if  for  the  kinetic  energy,  the  velocity  is  the  same, 
there  is  no  change;  but  whenever  there  is  a  difference  there 
will  follow  a  change  within  the  system.  The  third  principle 
of  thermodynamics  says  that  these  changes  always  follow  a 
path  which  requires  the  least  effort.  This  is  sometimes  named 
Hamilton's  principle.  With  these  theories  of  readjustment 
and  flux  of  energy  the  occasion  and  character  of  the  various 
changes  or  phenomena  of  the  material  world  may  be 
schematized. 

It  is  worthy  to  note  that  no  one  has  succeeded  in  exactly 
and  completely  reversing  a  series  of  natural  processes.  There 
is  always  a  loss  of  energy  usually  as  heat,  in  any  series  of 
transferences  or  transformations  of  energy.  The  researches 
of  Clausius  and  Planck  seem  to  prove  that  there  is  a  constant 
"degradation"  of  energy  or  a  reduction  to  the  condition  of  a 
dead  level.  Without  tension  or  difference  in  potential  there 
is  no  transmission  of  energy,  nor  can  there  be  any  work  done. 

Having  attained  then,  the  mechanical  conception  of  energy 
and  the  principles  of  conservation,  we  come  into  possession  of 
a  unified  theory  and  a  workable  scheme  of  antecedents  and 
sequences  of  the  gross  phenomena  of  nature,  which  now  become 
a  subject  of  calculation  by  mathematical  analysis  as  formulated 
in  Analytical  and  Celestial  Mechanics. 

Granted  a  certain  quantity  of  energy  in  a  material  system, 


THE   MODERN    PERIOD.  125 

the  conditions  of  its  transfer  and  transformation  are  now  be- 
come a  matter  of  mathematical  calculation,  and  the  concomi- 
tant gross  phenomena  may  be  predicted  with  certainty  and 
precision.  The  great  principle  of  conservation  of  energy  is  a 
wider  generalization  than  the  Newtonian  mechanics.  It  has 
enabled  us  to  advance  our  explanation  of  the  motion-phenom- 
ena of  the  universe,  but  we  are  still  far  from  explaining  all 
phenomena  by  Mechanics.  The  result  of  recent  efforts  to  ex- 
tend the  science  so  as  to  explain  the  minuter  and  more  subtle 
phenomena  of  the  universe  will  now  be  briefly  commented 
upon. 

6.  The  Ether.    Energy.    Dissociation  of  Matter. 

The  nineteenth  century  saw  the  general  acceptance  of 
Lavoisier's  adage,  "Nothing  is  created,  nothing  is  lost."  With 
the  gradual  establishment  of  the  idea  of  conservation  came  an 
enthusiastic  endeavor  to  unite  the  various  separate  sciences 
into  Science  by  means  of  the  concept  of  energy.  Energy  being 
conceived  as  a  measure  of  activity  and  the  quantity  of  energy 
being  considered  invariable,  it  is  logical  to  expect  that  all  the 
phenomena  of  the  universe  might  be  co-ordinated  by  this  idea. 

Mechanics  which  had  developed  the  concept  of  energy  and 
a  series  of  mathematical  equations  expressing  its  relations, 
from  a  study  of  the  gross  motion  phenomena  of  the  world, 
had  arrived  at  what  appeared  to  be  a  universal  law.  And 
now  the  various  separate  chains  of  phenomena  which  had 
been  linked  together  by  the  chemist,  the  physicist,  the  botanist 
and  the  biologist  were  to  be  welded  into  one  Science  by  the 
principles  of  mechanics.  The  chemists  had  been  working 
toward  the  idea  of  conservation  for  nearly  a  century  and 
when  chemistry  and  mechanics  came  into  accord  upon  the 
idea  of  conservation,  it  was  felt  that  it  must  fit  the  other 
sciences  too,  and  that  it  was  the  key  to  nature's  secrets. 

A  review  of  the  scientific  beliefs  of  twenty-five  years  ago 
reveals  a  faith  in  the  duality  of  natural  phenomena.  They 
were  conceived  as  the  result  of  the  action  of  indestructible 
energy  through  indestructible  matter  which  was  conceived  as 
floating  in  an  all  pervading  medium  called  the  ether  of  space. 


126  THE   SCIENCE   OF   MECHANICS. 

This  medium  was  conceived  as  penetrating  and  pervading  all 
matter. 

The  idea  of  an  ether  of  space  appears  to  be  very  old.  The 
term  is  derived  from  the  Greek  word  aether,  meaning  the 
brilliant  upper  air.  The  hypothesis  in  later  times  was  the 
result  of  the  logic  that  demanded  a  medium  to  transmit  light 
and  heat  through  interplanetary  space  and  through  a  vacuum. 
Hence  it  was  at  first  called  the  light-bearing  or  luminiferous 
ether. 

Fresnel  (i 788-1 827),  the  French  physicist,  in  his  undulatory 
theory  of  light  first  gave  this  hypothesis  definition.  Later 
Faraday  (1791-1867)  likewise  postulated  a  medium  in  connec- 
tion with  his  researches  in  electricity  and  magnetism  and 
suggested  that  perhaps  one  and  the  same  medium  would  serve 
for  both  light  and  electricity.  The  researches  and  calculations 
of  numerous  investigators  among  whom  Maxwell  was  promi- 
nent finally  gave  decision  in  favor  of  one  medium  or  ether, 
possessing  certain  characteristics. 

Being  a  purely  arbitrary  hypothesis  the  ether  could  and 
soon  came  to  be  endowed  with  such  properties  as  were  called 
for  by  the  logic  of  the  situation,  and  these  properties  were 
altered  from  time  to  time  as  seemed  necessary.  The  ether 
was  declared  to  possess  inertia,  because  time  was  required 
for  the  propagation  of  light  through  it.  It  was  conceived  as 
having  density  and  elasticity  by  analogy  with  matter,  and  it 
was  pictured  as  an  "elastic  jelly."  In  this  medium,  waves 
varying  in  length  from  miles  to  less  than  two  millionths  of  a 
millimeter  were  conceived  as  explaining  various  phenomena  of 
light,  heat,  electricity  and  magnetism.  Though  nothing  is 
positively  known  of  the  existence  or  structure  of  the  ether, 
this  convenient  assumption  has  been  developed  with  great 
definiteness. 

Once  this  hypothesis  was  established  Mechanics  entered 
upon  a  new  phase  of  development.  It  was  called  upon  to 
deal  with  molecular  and  atomic  energy  and  invited  to  explain 
by  its  principles  the  minute  phenomena  of  light,  electricity  and 
biology.  In  this  it  relied  upon  the  unifying  power  of  the  law 
of  conservation  and  the  license  to  warp  and  model  the  sup- 
posititious ether  to  the  exigencies  of  the  occasion. 


THE   MODERN   PERIOD.  127 

How  far  this  has  been  successful  can  be  but  briefly  considered 
here.  It  soon  became  apparent  that  the  molecule  or  smallest 
portion  of  physical  matter,  sometimes  pictured  as  bearing  to 
a  drop  of  water  the  ratio  that  a  golf  ball  bears  to  the  earth, 
must  give  up  its  simplicity  as  a  dense  hard  sphere  and  become 
constituted  of  at  least  several  atoms  of  various  densities  to 
comply  with  the  chemist's  notions  of  elementary  and  com- 
pound substances. 

Before  long,  these  atoms  had  assumed  the  complexity  of 
solar  systems  and  were  conceived  as  composed  of  thousands 
of  particles  or  electrons  in  rapid  motion,  and  as  being  of  many 
varieties.  Here  we  see  at  work  the  familiar  old  primitive  no- 
tions of  division,  moving  particles  and  pictorial  representation. 
In  the  hands  of  such  investigators  as  Fizeau,  Crookes,  Kelvin, 
Lodge,  Le  Bon,  Michelson,  Morley,  Rayleigh,  Ramsey,  Roent- 
gen, J.  J.  Thomson,  Rutherford  and  others,  the  method  has 
been  applied  in  linking  up,  by  the  principles  of  gross  mechanics 
a  variety  of  minute  phenomena.  It  has  led  experimental  re- 
search through  numerous  novel  and  remarkable  investigations 
in  light,  heat  and  electricity  from  which  much  is  expected. 

With  the  discovery  of  the  X-Rays  by  Roentgen  in  1895,  and 
of  radioactivity  by  Becquerel  and  the  Curies  in  1898,  and  with 
the  discovery  by  J.  J.  Thomson  that  the  passage  of  these 
activities  through  the  air  makes  it  a  conductor  of  electricity, 
new  conceptions  arise.  The  air  as  we  commonly  know  it,  is 
a  non-conductor  of  electricity  but  "ionized"  air  produced  by 
radioactivity,  or  by  the  emanations  from  such  substances  as 
radium,  thorium,  and  polonium,  is  a  conductor.  It  soon  be- 
came evident  that  a  great  many  bodies  in  nature  are  spon- 
taneously active  and  are  constantly  giving  out  emanations. 

Investigation  showed  that  these  emanations  have  the  power 
of  dissociating  a  gas,  or  of  breaking  it  up  into  particles,  com- 
parable with  hydrogen  atoms,  and  particles  approximately  one 
thousandth  as  large,  called  electrons.  The  velocity  of  these 
particles  approximates  that  of  light  and  their  total  mass  or 
inertia  appears  to  be  due  to  an  electric  charge  in  motion.  In 
other  words  the  one  characteristic  invariable  property  of  mat- 
ter, viz:  mass,  is  explained  as  an  electric  charge  in  motion. 


128  THE   SCIENCE   OF  MECHANICS. 

Larmor,  in  his  "Ether  and  Matter,"  says  the  atom  of  matter  is 
composed  of  electrons  and  of  nothing  else.  This  conception 
builds  matter  of  electricity  in  motion,  though  it  is  a  question  as 
to  whether  this  is  a  simplification  or  a  complication  of  theory. 

The  question  as  to  where  these  electrons  get  their  motion, 
or  what  is  the  origin  of  the  energy  which  expels  these  emana- 
tions with  such  terrific  velocity,  has  been  met  by  a  mechanical 
hypothesis  of  the  atoms  as  whirling  "solar  systems"  of  thou- 
sands of  electron-satellites,  some  of  which,  when  equilibrium  is 
disturbed,  fly  off  tangentially  with  great  velocities.  This  is 
practically  saying  that  molecules  and  atoms  of  matter  on  their 
disruption  or  dissociation  set  free  energy.  Experiments  on 
radioactivity  show  that  a  gram  of  radium  will  raise  the  tem- 
perature of  100  grams  of  water  i°  C.  an  hour  without  per- 
ceptible loss  of  weight  on  the  chemist's  balance.  But  the  re- 
searches of  Prof.  Crookes  and  Dr.  Heydweiller,^  estimate  the 
duration  of  a  gram  of  radium  at  about  lOO  years  after  which 
there  is  no  longer  any  radium,  therefore  a  quantity  of  highly 
heated  water  may  be  left  as  a  result  of  its  emanations  if  we 
conceive  it  to  act  upon  water.  Here  matter  disappears  and 
energy  in  the  form  of  steam  pressure  appears  in  exact  ratio. 

This  brings  us  face  to  face  with  a  contradiction  of  the  law 
of  conservation  as  we  have  stated  it.  We  have  matter  fading 
into  the  ghost  of  matter  losing  its  one  distinguishing  unalter- 
able characteristic,  namely,  mass,  and  liberating  an  enormous 
quantity  of  energy  in  the  process.  From  a  mechanical  point 
of  view  this  is  a  contradiction  in  terms  but  the  advance  guard 
on  the  skirmish  line  of  science  necessarily  uses  the  terms  that 
are  at  hand  with  various  mental  reservations  and  modifications 
until  nomenclature  can  be  revised  and  remodeled.  With  every 
advance  in  Science  there  is  inevitably  a  period  of  temporary 
anarchy  in  theory  and  terminology.  The  concepts  of  energy 
and  electricity  appear  to  be  about  to  go  through  some  such 
period  of  transformation  as  has  happened  with  the  term  force. 

We  find  ourselves  now  on  the  threshold  of  the  realization 
of  the  dream  of  the  alchemist.     These  X-rays,  emanations, 

^P.  237. 

'Phys.  Zeitschrift,  October  15,  1903. 


THE    MODERN   PERIOD.  1 29 

ions,  electrons  and  electricity  appear  to  be  phases  of  the 
dematerialization  of  matter,  stages  in  the  breaking  down  of 
matter  into  intra-atomic  energy.  As  Professor  de  Heen  of 
Liege  says,  "it  seems  we  find  ourselves  confronted  by  condi- 
tions which  remove  themselves  from  matter  by  successive 
stages  of  cathode  and  X-ray  emissions  and  approach  the  sub- 
stance designated  as  the  ether." 

Further  researches  indicate  that  electricity  is  one  of  the 
forms  of  energy  that  result  from  the  breaking  up  of  atoms, 
that  it  is  composed  of  these  imponderable  electrons,  the  ghostly 
emanations  of  fading  matter  which  themselves  have  been  pic- 
tured as  but  minute  whirls  in  the  all  pervasive  ether.  We  come 
here  to  a  new  conception,  matter  is  conceived  as  built  up  of 
electrons,  pictured  as  little  whirl-pools  in  a  fundamental  ether 
of  which  the  universe  is  composed.  However  this  may  be,  we 
are  made  acquainted  with  stores  of  energy  and  activities  as 
little  known  as  electricity  was  before  Volta's  day.  The  estab- 
lishment of  the  fact  of  the  dissociation  of  matter  opens  up 
unsuspected  and  inconceivable  sources  of  energy.  The  energy 
liberated  from  the  partial  dissociation  of  a  tub  of  water  would 
probably  equal  that  of  all  the  anthracite  coal  fields  of  America. 

This  theory  hints  at  an  explanation  of  some  of  the  mysterious 
activities  of  vegetable  and  animal  life.  The  researches  of  bio- 
logical chemistry  are  just  beginning  to  reveal  some  of  the 
secrets  of  the  flux  and  reflux  of  intra-atomic  energy  in  highly 
complicated  and  unstable  compounds  and  the  incidental  liber- 
ation of  (electrical)  energy.  The  theory  also  offers  suggestions 
as  to  the  character  of  allotropy,  catalytic  action,  diastases, 
toxins  and  protoplasmic  action.  These  minute  phenomena  of 
nature  are  motion-phenomena  and  as  such  come  within  the 
purview  of  mechanics,  but  in  the  development  of  a  theory  of 
the  grosser  phenomena  they  have  had  scant  attention.  It 
may  be  that  the  laws  of  gross  mechanics  do  not  apply  here 
exactly,  at  any  rate  it  seems  that  there  is  enough  suspicion  of 
mutation  of  matter  and  flow  of  energy  to  put  the  law  of  con- 
servation on  the  defensive. 

The  most  radical  contradiction  of  the  now  commonly  ac- 
cepted doctrine  of  conservation  is  that  given  by  Dr.  Gustave 


130  THE   SCIENCE   OF  MECHANICS. 

Le  Bon  in  his  "Evolution  of  Matter,"  1905,  from  which  the 
following  summary  is  taken. 

"i.  Matter,  hitherto  deemed  indestructible,  vanishes  slowly 
by  the  continuous  dissociation  of  its  component  atoms. 

"2.  The  products  of  the  dematerialization  of  matter  con- 
stitute substances  placed  by  their  properties  between  ponder- 
able bodies  and  the  unponderable  ether — that  is  to  say  between 
two  worlds  hitherto  considered  as  widely  separate. 

"3.  Matter,  formerly  regarded  as  inert  and  only  able  to 
give  back  the  energy  originally  applied  to  it,  is  on  the  other 
hand,  a  colossal  reservoir  of  energy — of  intra-atomic  energy 
— which  it  can  expend  without  borrowing  anything  from 
without. 

"4.  It  is  from  the  intra-atomic  energy,  manifested  during 
the  dissociation  of  matter  that  most  of  the  forces  in  the  uni- 
verse are  derived,  notably  electricity  and  solar  heat. 

"5.  Force  and  matter  are  two  different  forms  of  one  and 
the  same  thing.  Matter  represents  a  stable  form  of  intra- 
atomic  energy;  heat,  light,  electricity,  etc.,  represent  unstable 
forms  of  it. 

"6.  By  the  dissociation  of  atoms, — that  is  to  say,  by  the 
dematerialization  of  matter,  the  stable  form  of  energy  termed 
matter  is  simply  changed  into  those  unstable  forms  known  by 
the  names  electricity,  light,  heat,  etc. 

"7.  The  law  of  evolution  applicable  to  living  beings  is  also 
applicable  to  simple  bodies;  chemical  species  are  no  more  in- 
variable than  are  living  species." 

These  are  bold  generalizations  made  from  comparatively 
scanty  experimental  data  on  very  minute  and  delicate  phe- 
nomena, and  they  are  not  unchallenged.  But  they  suggest 
a  new  departure  and  a  new  phase  of  development  in  mechanics 
and  hint  at  marvels  until  now  undreamt  of. 

As  to  the  possibility  of  producing  energy  for  industrial 
purposes  by  breaking  down  or  using  up  matter  and  thus  turn- 
ing it  into  energy,  the  expectation  is  certainly  as  bright  as 
was  the  prospect,  that  Volta's  early  electrical  experiments 
with  frogs'  legs  and  a  copper  wire  would  ever  lead  to  the 
operation  of  heavy  railroad  trains  by  electricity  or  to  the 


THE   MODERN   PERIOD.  I3I 

transmission  of  the  voice  from  city  to  city,  by  wire,  or  of 
"wireless  messages"  from  mid-ocean  to  shore. 

The  wonders  of  aerial  telegraphy  and  telephony  are  the 
result  of  careful  investigation  and  study  in  this  new  field  of 
what  might  be  called  the  mechanics  of  the  ether.  When  the 
"activities  in  the  ether"  are  more  thoroughly  understood  we 
may  expect  greater  wonders.  It  is  to  be  noted  that  it  is  not 
always  the  most  intense  action  that  will  produce  a  desired 
result.  A  thunder  clap  will  not  move  a  tuning  fork  to  vibra- 
tion, whereas  the  vibration  of  a  violin  string  will  do  so  if  of 
the  proper  key.  A  spark  is  ridicuously  inadequate  as  com- 
pared with  the  explosion  of  energy  it  may  cause.  The  simple 
striking  of  a  phosphorus-match  by  moving  it  with  a  velocity 
of  about  ten  feet  a  second,  serves  to  set  up  disturbances 
which  have  a  velocity  of  186,000  miles  a  second. 

Atomic  energy,  of  the  existence  of  which  there  seems  to  be 
no  doubt,  is  practically  inexhaustible  in  amount,  as  simple 
calculations  show.  The  energy  that  would  flow  from  the  dis- 
sociation of  a  one  cent  copper  coin  is  equal  to  the  energy  of 
1,000  tons  of  coal  applied  in  the  production  of  steam.  Me- 
chanics has  brought  us  from  the  dim  gropings  of  the  Stone 
Age,  for  "more  power  to  the  arm,"  to  an  outlook  upon  an 
immense  universe  of  ceaseless  energy.  When  mechanical  con- 
trivance shall  have  caught  up  with,  and  exploited  this  vision 
we  may  expect  a  conquest  of  power  that  will  accomplish  incon- 
ceivable wonders. 

This  then  is  the  fruit  of  fifty  centuries  of  patient  endeavor 
in  mechanics,  of  2,000  years  of  geometrical  mechanics  and  200 
years  of  analytical  mechanics.  It  is  the  heritage  of  the  patient 
fidelity  and  stern  integrity  of  the  great  inquirers  and  their  nu- 
merous minor  coadjutors,  and  it  presages  a  greater  and  more 
marvellous  harvest  of  enlightenment  and  benefaction  for  the 
future.  The  indomitable  courage  and  patience  of  these 
searchers  for  ultimate  and  invariable  truth  have  emancipated 
the  race  from  much  of  the  incubus  of  the  superstitious  fetish- 
ism, and  from  some  of  the  drudgery  of  daily  life,  and  they  point 
prophetically  to  greater  conquests  to  come.     But  in  the  words 


132  THE   SCIENCE  OF  MECHANICS. 

of  one  of  the  eminent  sages^  of  the  science,  all  have  thus  far 
been  as  little  children  picking  pebbles  on  the  shore,  while  the 
great  ocean  of  the  unknown  glooms  beyond.  The  words  of 
Laplace  are  still  all  too  true,  "What  we  know  is  little,  what  we 
do  not  know,  immense." 

^Sir  Isaac  Newton. 


PART   IV. 
CONCLUSION. 

The  history  of  the  science  of  mechanics  has  now  been  traced 
in  outHne.  We  have  noted  its  aspirations;  we  must  now  note 
its  limitations.  Science  is  human  experience  tested  and  ar- 
ranged in  order.  It  is  not  its  purpose  to  offer  a  philosophy 
of  the  universe,  nor  is  it  essentially  in  conflict  with  religion. 
It  seeks,  rather  to  co-ordinate  experiences  into  a  systematic 
theory  of  relations,  of  causes  and  effects.  The  discovery  of 
natural  truths  and  the  extension  of  the  field  of  knowledge  by 
a  process  of  correlation,  rejection,  revision  and  verification  is 
its  province. 

We  note  that  the  science  is  a  mental  resume  of  the  growing 
experience  of  the  race,  a  development  founded  on  many  cen- 
turies of  endeavor  in  the  arts  and  trades.  It  had  its  origin 
in  the  dim  past  with  geometry  which  evolved  from  land- 
surveying  as  mechanics  did  from  the  trades.  The  science  is 
essentially  the  product  of  European  thought.  In  the  nature 
of  things  its  development  consisted  in  abstracting  from  the 
numerous  phenomena  of  nature  the  constant  elements,  this 
method  obviously  indicating  itself  as  the  path  of  progress. 
Once  the  abstractions  of  form  and  position  were  realized,  study 
of  forms  and  positions  led  to  the  development  of  a  geometry 
of  measurement  and  an  arithmetic.  Until  this  point  is  reached 
not  much  can  be  expected  in  physical  science,  for  the  spur  of 
progress  is  the  question  "how,"  and  no  satisfactory  answer 
can  be  given  to  it  until  a  system  of  measurements  is  developed. 

When  once  the  abstract  conceptions  of  form  and  position 
are  firmly  established  and  a  method  of  measurements  devised, 
then  the  conditions  and  circumstances  of  change  of  position 
and  of  change  of  form  and  size  present  themselves  as  questions 
of  possible  investigation. 

Even  after  the  Greeks  had  developed  geometry,  their  ideas 

133 


134 


THE   SCIENCE   OF  MECHANICS. 


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Q    . 


2  S 
■-TJ  o 

3  03 

3 


-^  >-   en 

5-^  2-a 


en    en   -3  r-- 

(L>   ej  ji   c3 

o  ^34:;  0)  h 


o. 


03 


3     . 

3    -o  «  2  b 

3        3-1    3   o   g 
^  ii    >..°  ^   O      . 

S   3  5   en         '^^.y-M   0-5   C^ 

+-'  o!  3  o    ..y  3  ts  -M  P  j3  °  "u 


2^  o  rt  ^a^^ 


03  "C    eu'e    t"   03   o3 
-3  J=  U    >.^  ^  g 


en   g^    3    >>Ti 

>=  3  r^  g  O  t, 

^  0)  ^  g  OJ  OS 
>p  3  03-^-3  a 
KPhhJWH 


pa  o 


0 

0 

0  0  0  «o 

0 

0 

0  0  0  N 

ID 

to 

VO  lO^N 

O  O   l>.e»   M 
O  O   t>.0    'i- 


N  CD  >0\0  ro 

\o  vo  *o  vo  r^ 

vD  vO  vO  ^O  vo 


<I^      11      Wt-C      hH      l-l      MI-HI-ll-lt-l 


CONCLUSION.  135 


O  o«  c<  1-1   ■*  1000  00        00  t^ 

7  7        7       TTTTT     TT 

11   -jj-  in  -rf^   0^  O   1-1         O   rO 


U  U 


^    3    -3    flj 

r/  o  c         .ti  p  oj  I-  S  c 


Mayer,  Joule, 
Helmholtz,  Tyn 
dall,  Kelvin, 
Maxwell. 

Crookes,  Becquer 
el,  Thomson,  th 
Curies,  Lodge, 
LeBon,  and 
others. 

0 

.s 

'0 

c 
0 

in 
'0 

Pi 

Ph 

a    .3     .2 


ON         'ii 


.S*  12     "aS  U 


cr 


,C5       3  StiS'O-S      4->tj       o       c       c 

Sis  z     <jeKH  ^s  s  w  s 


U 


VOOO  00  O-HiO't^        ■^oo 

^  vO  vO  t^  t^  t^  t^  t^        1^  (^ 


w 


w 


136  THE   SCIENCE   OF  MECHANICS. 

on  natural  phenomena  are  found  expressed  in  such  naive  state- 
ments as  those  of  Aristotle  to  the  effect  that  all  bodies  have  a 
place  and  seek  their  place;  bodies  float  because  of  their  form; 
if  they  move,  their  motion  is  either  natural  or  violent;  heavy 
bodies  go  down  because  they  belong  down  under  the  lighter 
ones;  the  heavier  they  are,  the  farther  down  they  belong  and 
the  faster  they  move  to  get  there!  It  was  not  till  the  abstract 
concept  of  force  was  completely  attained  to,  eighteen  hundred 
years  later,  and  the  concomitant  circumstances  of  it  studied 
and  measured  that  these  ideas  gave  place  to  our  modern  gen- 
eralizations. 

The  measurement  of  the  constant  elements  in  natural  phe- 
nomena naturally  began  with  bodies  near  at  hand  and  at 
rest,  with  statics  and  hydrostatics.  We  find  that  the  great 
contribution  of  Archimedes  is  not  so  much  the  rules  and 
methods  commonly  associated  with  his  name,  as  the  develop- 
ment of  a  science  of  measurement  and  its  application  in  the 
study  of  natural  phenomena.  With  this  the  science  of  me- 
chanics began. 

Here  at  its  very  basis  we  find  the  abstract  concept  and  the 
mathematical  notion  of  relativity  as  fundamental  to  the 
science.  Those  who  ask  a  physics  or  mechanics  without  ab- 
stractions and  mathematics  are  seeking  science  without  its 
most  essential  and  useful  features.  The  further  review  of 
the  science  indicates  that  an  evolution  of  abstract  concepts 
and  a  development  of  the  application  of  analysis  in  connection 
with  them  is  an  inevitable  necessity  of  its  progress. 

It  is  not  the  purpose  here  to  consider  metaphysical  and 
psychological  questions,  therefore  we  will  not  speculate  upon 
the  probability  or  possibility  of  developing  a  science  of  me- 
chanics on  another  basis  than  the  abstract  concept  linked  up 
mathematically.  That  we  cannot  know  matter  or  the  phe- 
nomena of  nature  except  as  mental  percepts  or  concepts  is  a 
trite  saying;  it  seems  to  follow  logically  therefore  that  our 
mechanics  must  be  built  up  of  these  elements,  however  much 
experience  and  study  may  change  and  elaborate  them. 

We  have  seen  how  the  notion  of  force  developed  and 
changed.     How  at  first  it  was  probably  anthropomorphic,  con- 


CONCLUSION.  137 

ceived  as  the  muscular  vigor  of  an  invisible  demi-god,  how  it 
underwent  transformation,  served  a  very  useful  purpose  in 
the  development  of  mechanics  and  is  now  discarded  in  some 
text-books  as  an  outworn  subjective  idea  to  which  there  is 
nothing  in  nature  to  correspond.  It  is  so  with  other  ideas, 
and  perhaps  many  of  them  will  continue  to  go  through  some 
such  development  and  transformation.  They  will  be  changed, 
modified  or  discarded.  Mechanics  was  developed  by  geo- 
metric methods  of  measurement  up  to  the  year  1700;  then, 
the  method  of  fluxions  and  limits  made  possible  the  investi- 
gation of  quantities  whose  value  is  continually  changing,  and 
the  science  made  a  wonderful  advance. 

In  all  mechanical  experience  there  are  two  conceptions,  how- 
ever, which  are  constantly  present,  and  form,  as  it  were,  the 
background — Time  and  Space.  They  appear  to  be  funda- 
mental and  ultimate;  being  irreducible,  they  cannot  be  com- 
pared with  anything,  and  are  indefinable.  With  them  a  third 
conception  variously  pictured  and  called,  matter,  energy,  ether 
or  electron,  suffices  to  form  the  rational  resume  of  phenomena 
which  is  called  mechanics.  The  riddle  of  time  and  space  is  a 
question  of  metaphysics,  not  of  mechanics.  Considering  it 
briefly  we  note  there  are  several  points  of  view.  One  may 
hold  after  the  manner  of  Trendelenberg  in  his  "Logische  Un- 
tersuchungen,"  Chap.  V,  that  there  exists  the  conception  of 
motion  quite  apart  from  the  ideas  of  space  and  time,  which 
he  derives  from  it.  His  endeavors  to  prove  a  knowledge  of 
motion  prior  to  any  idea  of  position  or  of  sequence  are  not 
convincing.  Though  the  theory  is  plausible,  it  is  not  proven. 
A  second  theory  due  largely  to  the  philosophy  of  Kant  and 
Hume  would  make  time  and  space  merely  modes  of  perception, 
ways  in  which  the  perceptive  faculty  distinguishes  objects. 
Though  this  is  not  admitted  generally  as  expressing  the  full 
truth,  it  indicates  clearly  the  intimate  relation  between  time 
and  space. 

They  are  bound  together  in  a  way  which  may  be  pictured 
by  supposing  space  to  represent  the  breadth  of  our  field  of 
perception,  then  time  would  represent  its  length.  Space 
marks  the  co-existence  of  perceptions  at  a  point  in  time,  so 


138  THE   SCIENCE   OF   MECHANICS. 

time  marks  the  progression  of  perceptions  at  a  position  in 
space.  The  two  modes  combined  give  us  motion  as  a  funda- 
mental way  in  which  we  conceive  phenomena. 

If  we  admit  this,  the  mode  of  perceiving  things  in  this  way 
would  seem  an  essential  feature  of  our  conscious  life.  With 
only  the  one  space  mode  of  perception,  we  could  know  only 
of  co-existing  things,  of  number,  position  and  measurement; 
our  science  would  be  limited  to  arithmetic,  algebra  and  geo- 
metry, and  the  phenomena  of  motion  would  not  exist  for  us. 
We  could  not  conceive  of  warmth,  weight,  hardness,  etc.,  for 
these  depend  upon  sequence,  on  time.  On  this  theory  the 
perceptive  faculty  sorts  sense  impressions  by  these  two  modes, 
as  upon  a  rack  or  frame-work;  and  if  the  simile  be  permitted 
one  may  say  both  co-ordinates  are  necessary.  Neither  infinity 
of  space  or  of  time,  or  empty  time  find  place  on  the  frame- 
work, nor  have  they  meaning  in  the  field  of  perception. 

Space  and  time  in  this  view  are  modes  of  perceiving  things. 
They  are  not  necessarily,  per  se,  infinitely  large  or  infinitely 
divisible,  but  are  essentially  relative.  The  reality  of  time 
and  space  is  not  a  point  of  discussion  in  this  paper,  but  this 
philosophical  theory  is  often  involved  with  the  philosophy 
that  denies  or  is  agnostic  as  to  the  reality  of  matter, — re- 
garding sense  perceptions  under  the  modes  of  time  and  space 
as  the  only  realitites. 

A  more  rational  point  of  view  and  one  that  will  appeal  to  a 
greater  number  is  the  theory  that  neither  denies  the  existence 
of  matter  and  motion  apart  from  perception  nor  affirms  that 
time  and  space  are  but  modes  of  thought,  and  that  motion 
is  the  concomitant  of  their  relation  to  each  other.  According 
to  this  theory  the  reality  of  moving  bodies  is  not  denied,  but 
it  recognizes  that  our  knowledge  of  them  is  limited  to  what 
we  may  perceive  and  conceive  of  under  the  modes  or  limita- 
tions of  time  and  space.  It  recognizes  our  hypotheses  and 
principles  as  approximations,  and  questions  whether  we  can 
ever  fully  conceive  of  or  comprehend  the  realities  of  nature 
in  all  their  completeness. 

The  concepts  developed  with  and  within  these  modes  of  time 
and  space,  such  as  geometrical  surface,  atom,  molecule,  force, 


CONCLUSION.  139 

etc.,  are  not  necessarily  the  realities.  Often  they  are  obviously 
not  so.  But  these  terms  are  very  useful  in  picturing  the  correla- 
tion and  sequence  of  phenomena.  The  continual  shifting  and 
the  evolution  which  we  see  going  on  in  our  scientific  concepts 
indicate  that  they  are  approximations,  and  points  the  necessity 
for  caution  in  speaking  of  them  as  realities,  or  of  projecting 
ideal  dancing  or  whirling  molecules  into  the  world  of  the  actual. 
As  to  this  third  conception  which,  together  with  time  and 
space,  suffices  to  illustrate  the  phenomena  of  nature,  it  is 
variously  conceived  and  described  as  matter,  as  ether,  and 
as  electrons.  The  first  and  oldest  of  these  was  the  idea  out 
of  which  the  others  have  arisen  as  later  and  broader  knowledge 
imposed  more  precise  requirements.  The  older  books  on 
mechanics  went  along  with  "corpus"  and  "moles"  until  the 
distinction  between  weight  and  mass  was  made.  Then 
"masses"  sufficed  until  more  recent  times  when  matter  came 
to  be  commonly  defined  in  the  text-book  by  its  properties 
of  extension  and  inertia.  The  modern  view  is  given  in  the 
little  book  "Matter  and  Motion,"  by  Clerk  Maxwell  thus, 
(page  163):  "All  that  we  know  about  matter  relates  to  the 
series  of  phenomena  in  which  energy  is  transferred  from  one 
portion  of  matter  to  another  till  in  some  part  of  the  series  our 
bodies  are  affected,  and  we  become  conscious  of  a  sensation. 
We  are  acquainted  with  matter  only  as  that  which  may  have 
energy  communicated  to  it  from  other  matter.  Energy,  on 
the  other  hand,  we  know  only  as  that  which  in  all  natural 
phenomena  is  continually  passing  from  one  portion  of  matter 
to  another.     It  cannot  exist  except  in  connection  with  matter." 

In  effect,  this  paragraph  defines  this  third  thing  as  a  medium 
for  the  storage  and  communication  of  energy,  without  telling 
what  it  is  or  what  energy  is.  Having  arrived  at  "energy"  as 
a  convenient  conception  and  being  under  the  necessity  of  con- 
ceiving of  it  as  stored  and  transmitted,  the  idea  of  matter  is 
made  to  assist  to  that  end. 

In  the  well  known  Treatise  on  Natural  Philosophy  by  Thom- 
son and  Tait  we  read  (p.  207):  "We  cannot  of  course,  give 
a  definition  of  matter  which  will  satisfy  the  metaphysician, 
but  the  naturalist  may  be  content  to  know  matter  as  that 


I40  THE   SCIENCE   OF  MECHANICS. 

which  can  be  perceived  by  the  senses  or  as  that  which  can  be 
acted  upon,  or  can  exert  force."  This  definition,  Hke  the 
first,  is  of  a  dual  character  indicating  matter  as  the  medium  of 
the  action  of  force  instead  of  energy.  Either  of  these  defini- 
tions will  serve  for  a  development  of  mechanics  from  the  view 
of  energy  or  of  force.  With  either  of  these  premises  granted 
a  logical  mechanics  is  possible. 

If  we  consult  Tait's  "Properties  of  Matter"  (pp.  12-13  and 
pp.  287-91),  we  read:  "We  do  not  know,  and  are  probably 
incapable  of  discovering  what  matter  is,"  and,  "The  discovery 
of  the  ultimate  nature  of  matter  is  probably  beyond  the  range 
of  human  intelligence.'"  This  is  at  least  decisive,  but  some 
would  probably  say  it  is  unnecessarily  blunt  and  discouraging. 

The  idea  of  matter,  proving  inadequate  with  the  advance 
of  the  science,  and  the  application  of  molecular  mechanics 
in  the  study  of  light,  electricity  and  heat,  the  theory  of  the 
ether  as  a  perfect  fluid  medium  and  a  perfect-jelly  medium 
was  introduced.  The  jelly-theory  of  the  ether  has  undoubt- 
edly been  of  value  in  simplifying  many  of  our  views  of  physical 
phenomena,  but  not  being  entirely  satisfactory,  the  "vortex 
atom"  and  "vortex  ring"  in  the  ether  were  invented.  This 
was  followed  by  Kelvin's  "ether-squirt."  From  periodic  varia- 
tions of  the  rate  of  squirting  as  influenced  by  the  mutual 
action  of  groups  of  squirts,  he  was  able  to  picture  and  deduce 
many  of  the  phenomena  of  chemical  action,  cohesion,  light 
and  electro-magnetism. 

A  more  recent  theory  is  the  corpuscular  or  electron  theory 
as  expounded  by  J.  J.  Thomson  in  "The  Corpuscular  Theory 
of  Matter,"  1907,  which  supposes  that  "the  various  properties 
of  matter  may  be  regarded  as  arising  from  electrical  effects." 
On  page  2  of  this  volume  we  read:  "This  theory  supposes 
that  the  atom  is  made  up  of  positive  and  negative  electricity. 
A  distinctive  feature  of  the  theory — the  one  from  which  it 
derives  its  name — is  the  peculiar  way  in  which  the  negative 
electricity  occurs  both  in  the  atom  and  free  from  the  atom." 
He  supposes  that  the  negative  electricity  always  occurs  as 
exceedingly  fine  particles  called  corpuscles,  and  that  all  these 
corpuscles,  whenever  they  occur,  are  always  of  the  same  size 


CONCLUSION.  141 

and  always  "carry  the  same  quantity  of  electricity."  "What- 
ever may  prove  to  be  the  constitution  of  the  atom  we  have 
direct  experimental  proof  of  the  existence  of  these  corpuscles." 
This  theory  has  the  advantage  of  being  able  to  explain  elec- 
trical metallic  conduction.  It  explains  mechanical  inertia  as 
the  self  induction  of  an  electric  current,  and  mass  on  the  basis 
of  the  velocity  of  the  corpuscles.  On  this  theory  matter  is 
conceived  of,  as  in  part  at  least  identical  with  electricity,  and 
the  properties  of  matter  are  explained  as  electrical  effects. 
But  this  still  leaves  a  third  concept,  electricity  in  addition 
to  time  and  space.  It  is  interesting  to  note  that  in  the  exposi- 
tion of  this  theory,  analysis  and  the  elementary  mechanical 
concepts  of  velocity,  mass  and  energy,  etc.,  are  used  to  attain 
to  this  new  idea.  This  illustrates  the  evolution  of  new  con- 
cepts from  the  old,  as  the  path  of  advance  in  mechanics,  and 
in  all  science. 

Yet  even  these  closer  approximations  cannot  be  regarded 
as  realities.  They  indicate  mechanical  actions  which  may  be 
close  approximations  of  the  reality  but  there  is  no  ground  for 
calling  them  identities. 

Dr.  Ernst  Mach  says,^  "purely  mechanical  phenomena  do 
not  exist.  They  are  abstractions  made  either  intentionally  or 
from  necessity  for  facilitating  our  comprehension  of  things." 
Though  this  is  not  conclusively  established,  the  mechanical 
theory  of  nature  does  seem  artificial.  There  is  no  reason  for 
believing  that  an  actual  mechanism  of  atoms  and  molecules 
as  some  scientists  present,  is  at  the  bottom  of  nature.  In  fact, 
recent  researches  in  the  electron  idea  tend  to  indicate  that  this 
is  unlikely.  But  this  pictorial  mechanical  method  developed 
by  European  thought  is  a  highly  serviceable  and  valuable  ex- 
pedient for  generalizing  experiences,  teaching  them,  and  apply- 
ing them.     It  serves  a  most  useful  purpose  in  investigation. 

As  J.  J.  Thomson  says  in  his  "Corpuscular  Theory  of  Mat- 
ter,"^  "From  the  point  of  view  of  the  physicist,  a  theory  of 
matter  is  a  policy  rather  than  a  creed;  its  object  is  to  connect 
or  co-ordinate  apparently  diverse  phenomena,  and  above  all  to 

'"Mechanics,"  p.  404. 
*Page  I. 


142  THE   SCIENCE  OF  MECHANICS. 

suggest,  Stimulate  and  direct  experiment."  Mechanics  will  no 
doubt  continue  to  develop  in  this  way  in  the  future  as  in  the 
past.  As  Professor  Ziwet  says:^  "It  is  now  pretty  generally 
recognized  that  Newton's  laws  of  motion  including  his  defini- 
tion of  force  are  not  unalterable  laws  of  thought  but  merely 
arbitrary  postulates,  assumed  for  the  purpose  of  interpreting 
natural  phenomena  in  the  most  simple  and  adequate  manner. 
.  .  .  It  is  now  coming  to  be  recognized,  as  researches  are  made 
in  the  electron  theory,  that  the  abandonment  or  generalization 
of  the  older  mechanics  must  lead  to  a  more  general  mechanics. 
It  will  probably  be  non-Newtonian,  based  on  the  development  of 
the  electron  theory  including  Newton's  laws  as  a  special  case." 
Prof.  Pearson  says:^  "We  must  hope  to  ultimately  conceptual- 
ize an  ether,  from  the  simple  structure  of  which  several  of 
the  laws  of  motion  postulated  for  particles  of  gross  matter 
may  directly  flow.  .  .  .  The  customary  definitions  of  mass 
and  force,  as  well  as  Newton's  statements  of  the  laws  of 
motion,  abound  in  metaphysical  obscurities.  It  is  also  ques- 
tionable whether  the  principles  involved  in  the  current  state- 
ments as  to  superposition  and  combination  of  forces  are  scien- 
tifically correct  when  applied  to  atoms  and  molecules.  The 
hope  for  future  progress  lies  in  clearer  conceptions  of  the  nature 
of  ether  and  of  the  structure  of  gross  matter." 

The  history  of  mechanics  in  the  past  indicates  that  before 
each  step  in  advance,  there  is  a  period  of  readjustment  and 
of  assimilation  of  previous  ideas.  We  appear  to  be  passing 
through  such  a  period  now.  The  great  generalizations  of  the 
law  of  gravitation  and  of  the  principles  of  conservation  of 
matter  and  of  energy  have  about  done  their  work  of  readjust- 
ment, and  have  been  assimilated  to  the  previous  ideas,  forming 
a  body  of  doctrine  as  a  basis  for  further  progress. 

As  indications  of  where  this  advance  may  be  expected,  one 
may  look  toward  the  points  at  which  investigators  are  dis- 
satisfied with  the  science,  or  are  not  in  accord  with  each  other. 
These  are  in  the  direction  of  the  ultimate  character  of  this 
third  concept  variously  termed  matter-energy,  ether-squirt, 

^Science,  Vol.  XXIII,  p.  50. 

*" Grammar  of  Science,"  p.  321. 


CONCLUSION.  143 

vortex-ring,  electron,  etc.,  and  in  the  direction  of  the  law  of 
gravitation.  This  principle  never  has  fitted  in  well  with  the 
other  principles  of  mechanics.  It  is  found  unsatisfactory  in 
that  its  action  seems  to  be  different  in  kind.  It  is  a  convenient 
generalization,  but  there  is  no  explanation  of  how  the  pull  of 
one  body  is  conducted  across  space  or  what  conducts  it.  The 
principle  of  action  at  a  distance  is  not  satisfactory.  It  was 
not  satisfactory  to  Newton.  Progress  is  to  be  expected  in  this 
direction,  and  a  beginning  has  already  been  made  with  the 
Electron  Theory.  It  is  possible  that  a  more  general  law  may 
be  evolved  from  the  study  of  potential  as  expressed  in  the 
equation  of  Poisson. 

V^F  =   —  47rp. 

This  formula  appears  to  express  a  mode  of  conception  of 
natural  phenomena  which  is  ai^Qst_asultimate  as  the  time 
and  space  modes.  It  may  be  that  a  law  or  mode  of  conception 
may  be  evolved  that  will  include  all  three  modes  in  one. 
But  though  this  is  foreshadowed  in  analysis  it  is  not  possible 
yet,  to  state  such  a  law  in  words. 

As  some  of  the  fundamental  concepts  of  formal  mechanics, 
such  as  matter  and  the  law  of  gravitation,  are  not  beyond  criti- 
cism, later  advances  will  very  probably  reformulate  the  science. 
It  may  indeed  be  necessary  partially  to  tear  down  the  present 
system  and  build  it  anew  on  different  lines,  when  the  funda- 
mentals are  more  correctly  perceived  and  comprehended.  The 
science  originated  with,  and  developed  from  a  study  of  gross 
bodies  with  motions  of  considerable  amplitude,  and  the  notions 
thus  obtained  have  been  refined  and  applied  in  picturing  the 
unseen.  The  minute  operations  which  produce  the  large  ap- 
pearances may  in  the  future  be  pictured  as  of  a  different  kind 
and  order  from  the  gross  things.  While  the  endeavors  of 
the  German  professors  Hertz  and  Boltzmann  in  this  direction 
cannot  be  called  successful,  they  indicate  the  tendency.  We 
should  be  careful  not  to  let  prejudice  in  favor  of  present  ideas 
and  methods  hamper  progress  as  prejudices  have  done  in  the 
past. 

The  curious  disintegrating  effects  of  radium  and  uranium 


144  THE   SCIENCE   OF  MECHANICS. 

and  their  derivative  products,  each  with  a  characteristic  "rate 
of  decay,"  tend  to  weaken  the  notions  of  immutability  and 
conservation,  and  to  reinforce  the  idea  that  formal  mechanics 
at  best  gives  but  a  hazy  picture  of  the  realities  of  the  world. 
But  it  is  a  model  or  a  picture  that  can  be  improved  and  brought 
more  in  accord  with  a  wider  and  more  varied  number  of 
phenomena.  Though  imperfect,  its  value  and  utility  in  ap- 
plied science  and  engineering  is  marvellous.  Its  economic 
value  is  beyond  question,  and  is  indeed  the  reason  of  its  exist- 
ence, and  one  of  the  strong  incentives  to  its  improvement. 

The  "conceptual  shorthand,"  by  which  the  resume  of  phe- 
nomena is  made,  will  no  doubt  be  improved,  but  it  seems  im- 
possible that  the  fundamental  concepts  of  time  and  space 
shall  give  place.  And  the  third  idea  which  is  at  present  neces- 
sary for  a  formal  presentation  of  the  science  appears  to  contain 
an  ultimate  element  not  resolvable  into  these  other  two.  The 
future  may  evolve  from  electricity  or  energy  a  more  precise 
idea  of  this  third  fundamental  concept  and  make  clearer  its 
connections. 

At  present,  in  spite  of  the  fact  that  some  of  the  generaliza- 
tions recorded  under  the  head  of  mechanics  are  widely  appli- 
cable in  the  world  of  phenomena,  we  cannot  claim  that  the 
science  comprises  a  knowledge  of  the  foundations  of  the  world 
of  phenomena,  nor  indeed  a  true  picture  of  any  reality  of  the 
world.  The  most  that  may  be  legitimately  claimed  is  that  it 
gives  a  tentative  mental  resume,  as  Dr.  Mach  says,  an  "as- 
pect" of  the  world  of  phenomena  which  is  fairly  satisfactory 
and  prodigiously  useful  and  valuable. 

Withal  we  should  be  on  our  guard  lest  our  science  be  too 
much  with  us,  late  and  soon,  lest  we  come  to  reverence  these 
apparent  constancies  of  relation  and  these  serviceable  fancies 
too  highly.  When  in  the  glories  of  sunset  and  rainbow  one 
sees  and  thinks  of  nothing  but  molecules  and  refractions,  then 
truly  there  is  "little  we  see  in  nature  that  is  ours." 

The  historical  review  of  the  development  of  the  Science  indi- 
cates that  it  is  not  essentially  in  conflict  with  Philosophy  or 
Religion.  It  speculates  not  why,  but  asks  how,  and  it  is  only 
the  tyro  who  finds  it  incompatible  with  piety. 


CONCLUSION.  145 

While,  with  some,  a  mechanical  explanation  of  all  nature 
is  an  avowed  ideal,  the  scientist  who  ponders  the  world  of 
phenomena  with  an  open-mind,  cannot  but  be  impressed  with 
a  causal  activity  immanent  therein,  which  is  more  than  blind 
chance.  The  universe  is  more  than  a  fortuitous  concourse  of 
molecules.  The  more  we  study  it,  the  more  need  we  have  to 
predicate  as  a  cause  of  the  cosmos  as  a  whole,  and  of  its  cease- 
lessly varying  infinity  of  phenomena,  an  Immanence  of  Con- 
trol, incomprehensible  to  our  finite  mind. 


146  THE   SCIENCE  OF  MECHANICS. 


A  BRIEF  BIBLIOGRAPHY  OF  NOTEWORTHY  PUBLICATIONS 
ON  THE   SCIENCE  OF   MECHANICS. 

Archimedes.     Oxford  manuscript,  edition  1792.     German  translation  of 

Nizze,  1824. 
Arneth,  A.     Die  Geschichte  der  reinen  Mathematik.     Stuttgart,  1852. 
Arago,  F.  J.  D.     Collected  works,  Paris,  1857,  containing  an  account  of 

various  mathematicians  of  the  middle  ages  and  modern  times. 
Bayma.     Molecular  Mechanics. 

Bernoulli,  J.     Opera  Omnia,  Acta  Eruditorium,  1693. 
Bernoulli,  D.     Hydrodynamica,  1738. 

Baden-Powell,  Historical  View  of  the  Progress  of  Physical  Science. 
Ball,  J.  R.  W.     The  History  of  Mathematics. 
Bossut,  C.     Histoire  generale  des  mathematiques,  1810.     Cours  complet 

des  mathematique,  7  vols.,  1801,  etc. 
Cantor,  M.     Vorlesungen  iiber  die  Geschichte  der  Mathematik.     Leipzig. 
Cajori,  F.     A  history  of  Physics. 
Clausius.     Die  mechanische  Warmetheorie,  1876. 
Clifford,  W.  K.     Common  Sense  of  the  Exact  Sciences. 
Crookes,  Sir  Wm.     Radiant  Matter. 
D'Alembert.     Traite  de  Dynamique,  1743. 
Delambre,  J.  B.  J.     Histoire  de  I'Astronomie. 
Daniel,  A.     Principles  of  Physics. 

Draper,  J.  W.     Conflict  between  Religion  and  Science. 
Diihring.    Kritische  Geschichte  der  Mechanik. 
Duncan,  R.  K.     The  New  Knowledge. 
Euler.     Methodus,  Opera,  1744  (Leipzig,  1887). 
Fleming,  Jas.     Electronic  Theory,  Popular  Science  Monthly,  1902. 
Fournier.    The  Electron  Theory. 
Galileo.     Discorsi,  16  vols.,  Alberi,  Florence,  1856. 
Gibbs,  J.  W.     Statistical  Mechanics. 
Gow,  J.     Short  History  of  Greek  Mathematics. 
Gunther,  S.     Vermischte  Untersuchungen  zur  Geschichte  der   mathema- 

tischen  Wissenschaften.    Leipzig,  1876. 
Hamilton.     Quaternions. 

Hankel,  H.     Zur  Geschichte  der  Mathematik.    Leipzig,  1874. 
Heller,  A.     Geschichte  der  Physik.     Stuttgart,  1882. 
Helm.     Die  Lehre  von  der  Energie. 
Hertz.     Principien  der  Mechanik. 
Huygens.     Horologium  Oscillatorium;  Opera. 
Holman.     Matter,  Energy,  Force  and  Work. 
Jerons.     Principles  of  Science. 
Joule.     Scientific  Papers,  2  vols. 
Kaestner,  A.  G.     Geschichte  der  Mathematik. 
Kretschmer.     Die  physische  Erdkunde  im  Mittelalter. 
Kimball.     The  Physical  Properties  of  Gases. 
Lami.     Elemens  de  Mecanique,  1687. 
Laplace.     Mecanique,  Celeste,  1799. 


BIBLIOGRAPHY.  1 47 

Larmor,  Jos.     Ether  and  Matter,  1901. 

Lagrange.     Mecanique  Analytique,  1788. 

Lehmann.     Molecular  Physik  (Leipzig,  1889). 

Lodge,  O.     The  Ether  of  Space.     Pioneers  of  Science.     On   Electrons. 

The  Electrician,  1903. 
Love.     Theoretical  Mechanics,  1897. 
MacLaurin.     A  Complete  System  of  Fluxions. 
Marie,  M.     Histoire  des  Sciences  Math,  et  Phys.  (Paris,  1888). 
Mach,  E.     The  Science  of  Mechanics,  1893. 
Mariotte.     Traite  du  Mouvement  des  Eaux,  1666. 
Maspero.     The  Dawn  of  Civilization. 

Maxwell,  C.     Matter  and  Motion,  1892.     Theory  of  Heat,  1897. 
Meyer.     The  Kinetic  Theory  of  Gases  (London,  1899). 
Michie,  P.     Elements  of  Mechanics. 
Minchin,  G.  M.     Treatise  on  Statics. 
Mivart,  St.  G.     The  Groundwork  of  Science. 
Murchard,  F.  W.     Litteratur  der  math.  Wissenschaften, 
Newton,  L     Principia. 
Nichols,  E.  F.     Physics. 
Pascal.     Traite,  1662. 
Pearson,  K.     The  Grammar  of  Science. 

Planck.     Das  Princip  der  Erhaltung  der  Energie  (Leipzig,  1877). 
Poggendorff,     J.     C.     Biographisch-Literarisches     Handworterbuch     zur 

Geschichte  der  exacten  Wissenschaften.    Leipzig,  1863. 
Poisson,  3.  D.     Traite  de  Mecanique,  1840. 
Poinsot.     Elemens  de  Statique,  1877. 
Poncelat.     Cours  de  Mecanique. 
Poynting  &  Thomson.     Properties  of  Matter,  1901. 
Quetelet,  L.  A.  J.     Histoire  des  Sciences  mathematiques  et  Physiques  chez 

les  Beiges,  1864.     Bruxelles. 
Risteen.     Molecules  and  Molecular  Theory,  1895. 
Rosenthal,  G.  E.     Encyclopaedia  der  Mathematik.     Gotha,  1796. 
Routh.     Rigid  Dynamics,  1884.     Dynamics  of  a  Particle,  1898. 
Rowland,     Scientific  Papers.    Baltimore,  1902. 
Stallo,  J.  B,     Modern  Physics. 
Stevinus,  S.     Hypomnemata  Mathematica,  1608. 
Tait,  P.  C.     Recent  Advances  in  Physical  Science.     Newton's  Laws  of 

Motion.     Thermodynamics.     The  Properties  of  Matter,  etc. 
Thomson,  J.  J.     The  Corpuscular  Theory  of  Matter.     The  Application 

of  Dynamics  to  Physics  and  Chemistry,  etc. 
Thomson  &  Tait.     Treatise  on  Natural  Philosophy. 
Thomson,  W.,  Sir  (Kelvin).     Lectures  and  Addresses. 
Tylor.     Primitive  Culture. 
Todhunter,  L     A  History  of  the  Calculus  of  Variation  during  Nineteenth 

Century. 
Varignon.     Nouvelle  Mecanique,  1687. 
Wallis.     Mechanica  Sive  de  Motu,  1670. 

White,  A.  D.     A  History  of  the  Warfare  of  Science  and  Theology  in  Christ- 
endom, 2  vols. 
Wren,  C.     Lex  Natural  de  Collisione  Corporum,  1669. 
Whewell.     History  of  the  Inductive  Sciences. 


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